On Derivation of the Poisson–Boltzmann Equation

Abstract

Starting from the microscopic reduced Hartree–Fock equation, we derive the macroscopic linearized Poisson–Boltzmann equation for the electrostatic potential associated with the electron density.

1 Introduction

1.1 The Reduced Hartree–Fock Equation

The success of the Hartree–Fock and density functional theories in revealing the electronic structure of matter warrants their use as a starting point in the derivation of emergent macroscopic properties of quantum matter.

Here, one of the central problems is the derivation of macroscopic Maxwell’s equations in dielectrics. The first attack on such a derivation was made in the pioneering works of Cancès, Lewin and Stoltz and E and Lu and their collaborators [5,6,7,8,9, 15,16,17, 17]. These works deal with the reduced Hartree–Fock equation (REHF)¹ and the Kohn–Sham equation (KSE) of the density functional theory (DFT) at zero temperature. The first treatment of the positive temperature REHF was given by Levitt [23] (see also [13]).

In this paper, we consider the REHF at positive temperature, which is also a simplified DFT equation, and derive from it the linearized effective Poisson–Boltzmann equation of electrostatics, widely used in molecular and structural biology (see e.g. [19]).

For a positive temperature T and with the electron charge set to $e=-1$, REHF can be written in terms of the one-particle negative charge (or probability) density $\rho (x)$ of the electron (or generally any Fermi) gas, as

$$\begin{aligned}&\rho = {\text {den}}[f_T(h_{ \rho } - \mu )], \end{aligned}$$

(1.1)

where ${\text {den}}: A\rightarrow \rho _A$ is the map from operators, A, to functions $\rho _A(x):=A(x, x)$ (here A(x, y) stands for the integral kernel of an operator A), $f_T(\lambda )$ is the Fermi–Dirac distribution,

$$\begin{aligned} f_T(\lambda ):= f_{FD}(\lambda /T),\ \quad f_{FD}(\lambda ) = \frac{1}{e^\lambda +1} \, \end{aligned}$$

(1.2)

(due to the Fermi–Dirac statistic), $\mu$ is the chemical potential and $h_{\rho }$ is a self-adjoint one-particle Hamiltonian depending on the density $\rho$ (self-consistency). Since $h_{\rho }$ is self-adjoint the r.h.s. of (1.1) is well defined. Assuming the electrons are subject to an external potential due to a positive charge distribution $\kappa$ (say, due to positive ions), $h_{\rho }$ is given by

$$\begin{aligned} h_{\rho }:=-\Delta - v*(\kappa -\rho ) \, , \end{aligned}$$

(1.3)

where v is an inter-particle pair potential. It is taken to be the electrostatic potential, as specified below.

Let $L^2_{\mathrm{loc}}\equiv L^2_{\mathrm{loc}}(\mathbb {R}^d)$ denote the space of locally square integrable functions. We fix a lattice $\mathcal {L}\subset \mathbb {R}^d$ and let $L^2_{\mathrm{per}}$ be the space of $L^2_{\mathrm{loc}}$, periodic w.r.t. $\mathcal {L}$ functions. Finally, let $(L^2_{\mathrm{per}})^\perp$ be the orthogonal complement of the constant functions in $L^2_{\mathrm{per}}$. In what follows, we assume $\rho -\kappa \in L^2+(L^2_{\mathrm{per}})^\perp$ and v is the electrostatic potential, $v(x)= \frac{1}{4\pi |x|}$ in 3D, or, generally,

$$\begin{aligned} v*f=(-\Delta )^{-1}f, \end{aligned}$$

for $f\in L^2+(L^2_{\mathrm{per}})^\perp$, so that $\Delta ^{-1}$ is well-defined.² For $\rho$’s and $\kappa$’s specified above, the operator $h_{ \rho }$ is self-adjoint.

The positive temperature, reduced Hartree–Fock equation (1.1) will be abbreviated, with the view to readability, as the TREHF.

For $T=0$, function (1.2) becomes the characteristic function of the interval $(-\infty , 0)$ and Eq. (1.1) becomes just the REHF.

1.2 Electrostatic Potential

Due to the choice $v*f=(-\Delta )^{-1}f$, the electrostatic potential $\phi =v*(\kappa -\rho )$ satisfies the Poisson equation

$$\begin{aligned} -&\Delta \phi = (\kappa -\rho ). \end{aligned}$$

(1.4)

Plugging $\rho$ from (1.1) into this equation and taking $v*(\kappa -\rho )=\phi$ in (1.3), we find the equation for $\phi$

$$\begin{aligned} -\Delta \phi = (\kappa - {\text {den}}[f_T(h^{\phi }-\mu )]), \end{aligned}$$

(1.5)

where

$$\begin{aligned} h^{\phi } = -\Delta - \phi . \end{aligned}$$

(1.6)

We can recover $\rho$ from $\phi$ via Eq. (1.4) or the equation

$$\begin{aligned} \rho = {\text {den}}[f_T(h^{\phi } - \mu )]. \end{aligned}$$

(1.7)

Let $H^s$ and $H^s_{\mathrm{per}}$ be the Sobolev spaces corresponding to $L^2$ and $L^2_{\mathrm{per}}$. If $\kappa , \rho \in L^2 + L^2_{\mathrm{per}}$, $\phi \in H^2 + H^2_{\mathrm{per}}$ and $\phi$ and $\rho -\kappa$ satisfy (1.4), then

$$\begin{aligned}&\int _\Omega \rho _{\mathrm{per}}=\int _\Omega \kappa _{\mathrm{per}}, \end{aligned}$$

(1.8)

where $\Omega$ is an arbitrary fundamental cell of $\mathcal {L}$ and the subindex ‘per’ denotes the periodic part of the corresponding function ($\in L^2 + L^2_{\mathrm{per}}$). Indeed, let $\Lambda _n:=\cup _{\lambda \in \mathcal {L}_n}(\Omega +\lambda )$, where $\mathcal {L}_n:=\mathcal {L}\cap [-n, n]^d$. Integrating (1.4) over the domain $\Lambda _n$ and using the Stokes’ theorem, we find

$$\begin{aligned} \int _{\partial \Lambda _n}\nabla \phi =\int _{\Lambda _n} (\kappa -\rho ). \end{aligned}$$

Since $\lim _{n\rightarrow \infty } \frac{1}{|\Lambda _n|}\int _{\partial \Lambda _n}\nabla \phi =0$ and $\lim _{n\rightarrow \infty } \frac{1}{|\Lambda _n|}\int _{\Lambda _n}(\kappa -\rho )= \int _\Omega (\rho _\gamma - \kappa )_{\mathrm{per}}$, the last relation gives (1.8).

Equation (1.8) shows that $\rho -\kappa \in L^2+(L^2_{\mathrm{per}})^\perp$, i.e. it satisfies the conditions mentioned in the paragraph after (1.3).

Equation (1.8) determines the chemical potential $\mu$ and expresses the conservation of the charge per fundamental cell of $\mathcal {L}$. It is considered as the solvability condition and should be added to (1.1) in the periodic case.

In what follows we associate with a solution $\rho$ of (1.1) the electrostatic potential

$$\begin{aligned} \phi _\rho = (- \Delta )^{-1} (\kappa -\rho ) \, , \end{aligned}$$

(1.9)

and with a solution $\phi$ of Eq. (1.5), the charge density $\rho$ according to (1.4), or (1.7).

1.3 Relation to the TEHF and KSE

The key positive temperature HFE is given by

$$\begin{aligned}&\gamma =f_T(h_{\gamma } - \mu ), \end{aligned}$$

(1.10)

where $f_{T}(\lambda )$ is as above and, for an external charge distribution $\kappa$,

$$\begin{aligned} h_{\gamma }:=-\Delta - v*(\kappa - \rho _\gamma ) + ex(\gamma ) \, . \end{aligned}$$

(1.11)

Here, recall, $\rho _\gamma (x):=\gamma (x, x)$ and $v*f=(-\Delta )^{-1}f$, and $ex(\gamma )$ (the exchange term) is the operator with the integral kernel $ex(\gamma )(x, y):=-v(x-y)\gamma (x, y)$, where $\gamma (x, y)$ is the integral kernel of $\gamma$. Observing that $h_{\gamma }\big |_{ex(\gamma )=0}=h_{\rho _\gamma }$, where $h_{\rho }$ is given in (1.3), one sees that (1.10) with $ex(\gamma )=0$ implies the equation

$$\begin{aligned}&\gamma =f_T(h_{\rho _\gamma } - \mu ). \end{aligned}$$

(1.12)

Equation (1.12) is equivalent to Eq. (1.1). Indeed, applying the map ${\text {den}}$ to Eq. (1.12) gives (1.1). In the opposite direction, if $\rho$ solves (1.1), then the density operator

$$\begin{aligned} \gamma = f_T(h_\rho - \mu ),\end{aligned}$$

(1.13)

acting on $L^2(\mathbb {R}^d)$, solves (1.12). Thus, (1.12) is the TREHF in terms of the density operator $\gamma$.

By replacing $ex(\gamma )$ in (1.11) by a local exchange-correlation term $\text {xc}(\rho )$ and then applying, as above, the map ${\text {den}}$ to the resulting equation, one obtains the natural extension of the original Kohn–Sham equation to positive temperatures:

$$\begin{aligned}&\rho = {\text {den}}[f_T(h_{ \rho }^{\mathrm{KS}} - \mu )], \end{aligned}$$

(1.14)

$$\begin{aligned}&h_{\rho }^{\mathrm{KS}}:=-\Delta - v*(\kappa -\rho ) +\text {xc}(\rho ) \, . \end{aligned}$$

(1.15)

1.4 The Origin of the TEHF/TREHF Equations

As the TEHF and TREHF arise in the same way, in order to avoid repetitions, we consider here only the later.

Equation (1.12) originates from the static version

$$\begin{aligned}&[h_{\rho _\gamma }, \gamma ]=0 \end{aligned}$$

(1.16)

of the time dependent RHF equation (see e.g. [12] for a review)

$$\begin{aligned} \partial _t \gamma = i[h_{\rho _\gamma }, \gamma ] \,. \end{aligned}$$

(1.17)

Indeed, ignoring symmetries and accidental divergence, $\gamma$ solves (1.16) if and only if $\gamma$ solves $\gamma =f((h_{\rho _\gamma }-\mu )/T)$ for some reasonable function f. (The parameters T and $\mu$ are of no significance at this stage; they are introduced for future reference.)

The selection of f is done on physics grounds, either bringing the system in question in contact with a thermal reservoir at temperature T and the chemical potential $\mu$, or passing to the thermodynamic limit. This leads to Eq. (1.12).

As we discuss below, Eq. (1.12) is the Euler-Lagrange equation for the natural free energy.

Remark

If the particles in question were bosons, then $f_{FD}$ would be replaced by the Bose–Einstein distribution

$$\begin{aligned} f_{BE}(\lambda ) = \frac{1}{e^\lambda - 1} \, . \end{aligned}$$

(1.18)

1.5 Results

We are interested in the dielectric response in a medium subjected to a local deformation of the crystalline structure. To formulate our results we introduce some notation and definition.

In what follows, we assume that $d=3$ and let $\mathcal {L}$ be a (crystalline) Bravais lattice in $\mathbb {R}^3$. We also define the Hilbert space of $\mathcal {L}$-periodic functions

$$\begin{aligned} L^2_{\mathrm{per}}\equiv L^2_{\mathrm{per}}(\mathbb {R}^3) = \{ f \in L^2_{\mathrm{loc}}(\mathbb {R}^3) : f \text { is } \mathcal {L}\text {-periodic } \}, \end{aligned}$$

(1.19)

with the inner product $\langle f, g\rangle = \int _\Omega {\bar{f}} g$ and the norm $\Vert f\Vert _{L^2_{\mathrm{per}}}^2 = \int _\Omega |f|^2$ for some arbitrary fundamental domain $\Omega$ of $\mathcal {L}$. We denote by $H^s_{\mathrm{per}}\equiv H^s_{\mathrm{per}}(\mathbb {R}^3)$ and $\Vert \cdot \Vert _{H^s_{\mathrm{per}}}$ the associated Sobolev spaces and their norms, while the standard Sobolev spaces and their norms are denoted by $H^s\equiv H^s(\mathbb {R}^3)$ and $\Vert \cdot \Vert _{H^s}$.

Crystals We consider a background charge distribution, $\kappa (y)\equiv \kappa _{\mathrm{per}}(y)$, periodic with respect to the lattice $\mathcal {L}$ (crystal). Here y stands for the microscopic coordinate.

We think of $\mathcal {L}$ and $\kappa _{\mathrm{per}}$ as a crystal lattice and the ionic charge distribution of $\mathcal {L}$. An example of $\kappa _{\mathrm{per}}$ is

$$\begin{aligned} \kappa _{\mathrm{per}}(y) = \sum _{l \in \mathcal {L}} \kappa _a(y-l) \, , \end{aligned}$$

(1.20)

where $\kappa _a$ denotes an ionic (“atomic”) charge distribution.

Dielectrics Next, we describe a model of the (crystalline) dielectric.

Definition 1.1

We say that an $\mathcal {L}$-periodic background charge density $\kappa _{\mathrm{per}} \in L^2_{\mathrm{per}}$ is dielectric, if TREHF (1.1), with $\kappa =\kappa _{\mathrm{per}}$, has an $\mathcal {L}$-periodic solution $(\rho _{\mathrm{per}}, \mu _{\mathrm{per}})$, with the following properties:

(a) the periodic one-particle Schrödinger operator

$$\begin{aligned}&h_{\mathrm{per}}:=h^{\phi _{\mathrm{per}}} =-\Delta - \phi _{\mathrm{per}}, \text { with} \end{aligned}$$

(1.21)

$$\begin{aligned}&\phi _{\mathrm{per}}:= 4\pi (-\Delta )^{-1}(\kappa _{\mathrm{per}}-\rho _{\mathrm{per}}), \end{aligned}$$

(1.22)

acting on $L^2 \equiv L^2(\mathbb {R}^3)$ is self-adjoint and has a gap in its spectrum;

(b) $\mu _{\mathrm{per}}$ is in this gap;

(c) $\phi _{\mathrm{per}} \in H^2_{\mathrm{per}}$ and $\Vert \phi _{\mathrm{per}}\Vert _{H^2_{\mathrm{per}}}$t$|\mu _{\mathrm{per}}|$$\le \Delta _{per}$, independently of T.

An existence result for the dielectrics is discussed in Remarks 6 and 7 after the next theorem. In particular, Proposition 1.3 shows that the set of dielectric charge densities $\kappa _{\mathrm{per}}$ is robust. Moreover, (1.5) can be reformulated so that only $\phi _{\mathrm{per}}$ and $\mu _{\mathrm{per}}$, but not $\kappa _{\mathrm{per}}$, enter it explicitly, see (1.44). So these are the only inputs of our analysis

Dielectric response We consider a macroscopically deformed microscopic crystal charge distribution,

$$\begin{aligned} \kappa _\delta (y) = \kappa _{\mathrm{per}} (y) + \delta ^{3}\kappa '(\delta y), \end{aligned}$$

(1.23)

where $\delta$ is a small parameter which stands for the ratio of microscopic and macroscopic scale and $\kappa '(x) \in L^2$ is a small local perturbation living on the macroscopic scale. By y and $x=\delta y$, we denote the microscopic and macroscopic coordinates, respectively. Thus, the microscopic scale is $y\sim 1$ and $x\sim \delta$ and the macroscopic one, $y\sim 1/\delta$ and $x\sim 1$.

We formulate the conditions for our main result. We introduce the homogeneous Sobolev spaces

$$\begin{aligned} \dot{H}^s\equiv \dot{H}^s(\mathbb {R}^3) = \left\{ f \text { measurable on } \mathbb {R}^3 : \Vert f\Vert _{\dot{H}^s} < \infty \right\} \end{aligned}$$

(1.24)

for $s\ge 0$ with the associated norm

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s}^2 = \int |(-\Delta )^{s/2} f(k)|^2 \,. \end{aligned}$$

(1.25)

[A1]:

(Dielectricity) $\kappa _{\mathrm{per}}$ is dielectric.

Let $h_{\mathrm{per}}$ and $h_{\mathrm{per, 0}}$ denote operators given by expression (1.21) acting on $L^2(\mathbb {R}^3)$ and $L^2_{\mathrm{per}}(\mathbb {R}^3)$, respectively. These operators are self-adjoint and the latter has a purely discrete spectrum. By Assumptions [A1], $\mu _{\mathrm{per}}$ is in a gap of $h_{\mathrm{per}}$. For notational convenience, we rescale our problem so that

$$\begin{aligned} \eta := \text {dist}(\mu _{\mathrm{per}}, \sigma (h_{\mathrm{per}})) =1. \end{aligned}$$

(1.26)

It follows the Bloch–Floquet decomposition results in Sect. 2.4 below that the gaps of $h_{\mathrm{per}}$ are contained in the resolvent set of $h_{\mathrm{per, 0}}$, so that

$$\begin{aligned}&\eta _0:= \text {dist}(\mu _{\mathrm{per}}, \sigma (h_{\mathrm{per}, 0}))\ge 1. \end{aligned}$$

(1.27)

• [A2] (Perturbation ${\kappa '}$)

  $$\begin{aligned} {\kappa '}\in H^1 \cap \dot{H}^{-1} . \end{aligned}$$

In what follows, the inequalities $A\lesssim B$ and $A > rsim B$ mean that there are constants C and c independent of T and $\delta$, s.t. $A\le C B$ and $A\ge c B$ and similarly for $A\ll B$ and $A\gg B$. Our main result is

Theorem 1.2

Let Assumptions [A1]–[A2] hold and let $(\phi _{\mathrm{per}}, \mu _{\mathrm{per}})$ be the electrostatic and chemical potentials associated with $\kappa _{\mathrm{per}}$ (entering [A1]) as per Definition 1.1.

There is $\alpha$$=\alpha (\Lambda _\mathrm{per})>0$ sufficiently small, s.t., if

• [A3] (Regime) The parameters $T > 0$ and $\delta >0$ satisfy

  $$\begin{aligned} c_{T}:=T^{-1} e^{-\eta _0/T} \le \alpha ,\ \quad c_{T}^{- 8/9} \delta \le \alpha , \end{aligned}$$

  (1.28)

then the following statements are true

• 1. Electrostatic TREHF (1.5), with $\kappa =\kappa _\delta$ given in (1.23) and $\mu = \mu _{\mathrm{per}}$, has a unique solution $\phi _\delta \in H^2_{\mathrm{per}} + H^1$;

• 2. The potential $\phi _\delta (y)$ is of the form

  $$\begin{aligned} \phi _\delta (y) = \phi _{\mathrm{per}}(y) + \delta \psi (\delta y) + \varphi _{\mathrm{rem}}(\delta y), \end{aligned}$$

  (1.29)

  where $\varphi _{\mathrm{rem}}(x) \in H^{1}$ and obeys the estimates (with $\dot{H}^0=L^2$)

  $$\begin{aligned} \Vert \varphi _{\mathrm{rem}}\Vert _{\dot{H}^i}\ll \alpha ^{\frac{1}{4}-\frac{1}{2} i} (c_T^{-1/2} \delta )^{2-i}, \end{aligned}$$

  (1.30)

  with $\alpha$ given in (1.28), and $\psi (x) \in H^1$ and satisfies the equation

  $$\begin{aligned} (\nu -\nabla \cdot \epsilon \nabla ) \psi = \kappa ', \end{aligned}$$

  (1.31)

  with a positive number $\nu >0$ and a constant real, symmetric $3 \times 3$ matrix, $\epsilon \ge 1- O(c_T^2)$;

• 3. $\epsilon \equiv \epsilon (T)$ and $\nu \equiv \nu (T, \delta )$ are given explicitly by (1.35)–(1.37) and (1.32)–(1.33), below.

We discuss Assumptions [A1] and [A3] in Remarks 6 and 10 and the statements of the theorem, in Remarks 1-4, below.

1.6 Discussion

• (1) Theorem 1.2(1) and Eq. (1.9) connecting the charge density $\rho$ with $\phi$ imply that RHF equation (1.1), with (1.23) and $\mu = \mu _{\mathrm{per}}$, has a unique solution $\rho _\delta \in L^2_{\mathrm{per}} + \dot{H}^{-1}$.

• (2) The quantity $\nu \equiv \nu (T, \delta )$ is defined as

  $$\begin{aligned}&\nu =\delta ^{- 2}|\Omega |^{-1}(m+ O(c_T^2)), \end{aligned}$$

  (1.32)
  $$\begin{aligned}&m = -\mathrm {Tr}_\Omega \left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] >0. \end{aligned}$$

  (1.33)

  Lemmas B.1 and B.2 of Appendix B imply the estimates

  $$\begin{aligned}&0<c_T\lesssim m \lesssim c_T. \end{aligned}$$

  (1.34)

  By (1.34) and (1.28), m is the leading term in (1.32) and $\nu \gg \delta ^{-7/8}$.

• (3) The $3 \times 3$ matrix, $\epsilon$, in (1.31) is given explicitly by

  $$\begin{aligned}&\epsilon := \mathbf {1}+\epsilon ' - \epsilon '', \end{aligned}$$

  (1.35)
  $$\begin{aligned}&\epsilon ' = -\frac{1}{|\Omega |}\mathrm {Tr}_{L^2_{\mathrm{per}}} \oint r_{\mathrm{per}, 0}^2(z) (-i\nabla ) r_{\mathrm{per}, 0}(z) (-i\nabla ) r_{\mathrm{per}, 0}(z) , \end{aligned}$$

  (1.36)
  $$\begin{aligned}&\epsilon '' = \frac{1}{|\Omega |} \langle \rho ',{\bar{K}}_{ 0}^{-1} \rho '\rangle _{L^2_{\mathrm{per}}} \, , \end{aligned}$$

  (1.37)

  where $r_{\mathrm{per}, 0}(z):=(z-h_{\mathrm{per}, 0})^{-1}$ and $h_{\mathrm{per}, 0}$ denotes the restriction of $h_{\mathrm{per}}:= h^{\phi _{\mathrm{per}}}= -\Delta +\phi _{\mathrm{per}}$ to $L^2_{\mathrm{per}}$, ${\bar{K}}_{ 0}$ is the operator defined in (4.11), and

  $$\begin{aligned} \rho ' = 2 {\text {den}}\oint r_{\mathrm{per}, 0}^2(z)(-i\nabla )r_{\mathrm{per}, 0}(z) \, . \end{aligned}$$

  (1.38)

• (4) Equations (1.31), (1.32) and (1.34) imply that

  $$\begin{aligned} \Vert \psi \Vert _{\dot{H}^{i}} =O([\delta |\Omega |^{1/2} m^{-1/2}]^{2-i}),\ i=0, 1, \end{aligned}$$

  and therefore, by (1.28), (1.30) and (1.34), we have

  $$\begin{aligned} \Vert \varphi _{\mathrm{rem}}\Vert _{L^2}\ll \alpha ^{1/4}(m^{-1/2} \delta )^2\ll \Vert \psi \Vert _{L^2}. \end{aligned}$$

  Hence $\psi$ is a subleading term in (1.29) in the $L^2$-norm.

• (5) (1.31) is the linearized Poisson–Boltzmann equation used extensively in physical chemistry and molecular biology (see e.g. [19]). $\epsilon$ is an effective permittivity matrix and $\sqrt{\nu }$ and $1/\sqrt{\nu }$ are the Debye-Hückel parameter and the Debye length, respectively.

The screening term $\nu$ in (1.31) is due to the electrons at the tail of the Fermi–Dirac distribution being at the conduction band. (In the macroscopic regime, the Fermi–Dirac distribution becomes the (Maxwell-) Boltzmann distribution.)

• (6) (Existence of crystalline dielectrics) We say that the potential $\phi$ is gapped if the Schrödinger operator $-\Delta - \phi$ has a gap in its continuous spectrum.

Proposition 1.3

For any $\mathcal {L}$-periodic, gapped potential $\phi _{\mathrm{per}} \in H^k_{\mathrm{per}}$, $k\ge 2,$ and any real number $\mu _{\mathrm{per}}$ in a gap of $h_{\mathrm{per}} := -\Delta - \phi _{\mathrm{per}}$, there is $\kappa _{\mathrm{per}} \in H^{k-2}_{\mathrm{per}}$ such that Eq. (1.1), with $\kappa =\kappa _{\mathrm{per}}$, has the solution $(\rho =\rho _{\mathrm{per}} \in H^{k-2}_{\mathrm{per}}, \mu =\mu _{\mathrm{per}})$ with the associated (according to (1.4)) electrostatic potential exactly $\phi _{\mathrm{per}}$. Moreover, the pair $(\phi _{\mathrm{per}}, \mu _{\mathrm{per}})$ satisfies the electrostatic Eq. (1.5) with this $\kappa _{\mathrm{per}}$.

Proof

Let $\phi _{\mathrm{per}}$ be such that $h_{\mathrm{per}} := -\Delta - \phi _{\mathrm{per}}$ has a gap. We choose $\mu _{\mathrm{per}}$ to be in this gap and define (see (1.6)–(1.7))

$$\begin{aligned} \rho _{\mathrm{per}} := {\text {den}}[f_T(h_{\mathrm{per}} - \mu _{\mathrm{per}})]. \end{aligned}$$

(1.39)

Next, we define

$$\begin{aligned} \kappa _{\mathrm{per}} := -\Delta \phi _{\mathrm{per}} + \rho _{\mathrm{per}}. \end{aligned}$$

(1.40)

Then, it is straightforward to check that $(\rho _{\mathrm{per}}, \mu _{\mathrm{per}})$ is a solution of Eq. (1.1) with background potential $\kappa _{\mathrm{per}}$. By construction, $h_{\mathrm{per}}$ has a gap and $\mu _{\mathrm{per}}$ is in this gap. $\square$

One can extend Proposition 1.3 to construct pairs $(h_{\mathrm{per}} = -\Delta - \phi _{\mathrm{per}}, \mu _{\mathrm{per}})$ having any desired property P. Following Proposition 1.3, we construct $\rho _{\mathrm{per}}$, $\phi _{\mathrm{per}}$, and $\kappa _{\mathrm{per}}$ via (1.39) and (1.40) in this order. Then $(\rho _{\mathrm{per}}, \mu _{\mathrm{per}})$ is a solution of Eq. (1.1) with background potential $\kappa _{\mathrm{per}}$. By construction, $h_{\mathrm{per}}$ has property P.

The proposition above shows that for any positive $\eta$ and T, we can find $\kappa _{\mathrm{per}} \in H^{k-2}_{\mathrm{per}}$ such that the solution of Eq. (1.1) with $\kappa =\kappa _{\mathrm{per}}$ and T gives the gap $\eta$.

• (7) (General dielectrics) We say that a background charge density $\kappa$ is dielectric if Eq. (1.1) with background charge distribution $\kappa$ has a solution $(\rho , \mu )$, with $\rho$ in an appropriate space, say, $H^2_{\mathrm{loc}}\cap L^\infty$, and having the following properties:

  • (a) the one-particle Schrödinger operator, defined for this solution,

    $$\begin{aligned}&h^\phi :=-\Delta - \phi , \text { with }\ \phi := 4\pi (-\Delta )^{-1}(\kappa -\rho ), \end{aligned}$$

    (1.41)

    acting on $L^2$, is self-adjoint and has a gap in its spectrum;

  • (b) $\mu$ is in this gap.

By the remark at the end of the previous item we have

Proposition 1.4

(Existence of general dielectrics) For any gapped potential $\phi \in H^2_{\mathrm{loc}}\cap L^\infty$ and any number $\mu$ in a gap of $h^{\phi } := -\Delta - \phi$, there is $\kappa \in L^2_{\mathrm{loc}}\cap L^\infty$ s.t. Eq. (1.1), with these $\kappa \in L^2_{\mathrm{loc}}\cap L^\infty$ and $\mu$, has the solution $\rho$, whose the electrostatic potential (according to (1.4)) is $\phi$.

• (8) (Existence of ideal crystals) The existence of periodic solutions to Eq. (1.1) (equilibrium crystalline structures exists at $T>0$) is shown in the following:

Theorem 1.5

Let $d=3$ and $\kappa _{\mathrm{per}} \in H^2_{\mathrm{per}}$. Then Eq. (1.1), with the $\mathcal {L}-$periodic background charge density $\kappa =\kappa _{\mathrm{per}}$ has a solution $(\rho _{\mathrm{per}}, \mu _{\mathrm{per}})$, with $\rho _{\mathrm{per}}$ periodic and satisfying $\sqrt{\rho _{\mathrm{per}}} \in H^1_{\mathrm{per}}$.

We give references to the proof of this theorem below.

• (9) In the limit $T \rightarrow 0$, our expression for the dielectric constant $\epsilon$ agrees with [7] (see Appendix A below).

• (10) (Physical dimensions) The physical cell size of common crystals is on the order of $10^{-10}$ ( [35]). This gives $\delta \sim 10^{-10}$. The gap size, $\eta _0$, is on the order 1eV [35]. Since the Boltzmann constant, $k_B$, is of the order $10^{-4} eV/K$, this gives $\eta _0/k_B\sim 10^{4} K$. Thus, though we do not compute actual constants in our estimate, we expect that the allowed values of $\delta$ and T are within physically interesting ranges.

• (11) (Energy) The evolution (1.17) conserves the number of particles $N_{X}(\gamma ):=\mathrm {Tr}_X(\gamma )$ and the energy

  $$\begin{aligned} E_{X}(\gamma )&:=\mathrm {Tr}_X\big ((-\Delta ) \gamma \big ) + \frac{1}{2} \int _X\sigma _\gamma v*\sigma _\gamma , \end{aligned}$$

  (1.42)

  where $X$ is either $\mathbb {R}^d$ or a fundamental cell $\Omega$ of $\mathcal {L}$, with $\mathrm {Tr}_X$ defined accordingly, and $\sigma _\gamma :=\kappa - \rho _\gamma$.

  Equation (1.12) is the Euler–Lagrange equation for the free energy functional

  $$\begin{aligned} F_{T}(\gamma ):=E_{X}(\gamma ) -T S_{X}(\gamma )-\mu N_{X}(\gamma ), \end{aligned}$$

  (1.43)

  where $S_{X}(\gamma ) = -\mathrm {Tr}_{\Omega } (\gamma \ln \gamma +(\mathbf {1}-\gamma ) \ln (\mathbf {1}-\gamma ))$ is the entropy.

To obtain the HF (free) energy functional, one should add to (1.42) ((1.43)) the HF exchange energy term $Ex(\gamma ):=\frac{1}{2} \int _X\int _X|v(x-y)\gamma (x, y)|^2$.

Literature The relation of the HF theory to the exact quantum many-body problem was established rigorously in [26].

For $T=0$, the existence theory for the RHFE and HFE was developed in [1, 11, 21, 26, 29], see [22, 25, 28], for reviews. For the Hartree–Fock equation (1.10) with periodic $\kappa = \kappa _{\mathrm{per}}$, the existence of periodic solutions from certain trace classes was obtained in [10] and [11].

Results for $T=0$, similar and related to Theorem 1.2, were proven in [5,6,7,8,9, 15,16,17].

For the case where $T > 0$, F. Nier [32] proved the existence and uniqueness of the TRHF (1.1) via variational techniques. Later, Prodan and Nordlander [33] provided another existence and uniqueness result with the exchange-correlation term in the case where $\kappa = \kappa _{\mathrm{per}}$ is small. In this case, the associated potential term $\phi _{\mathrm{per}} + \text {xc}(\rho _{\mathrm{per}})$, where $\text {xc}(\rho )$ is a local exchange-correlation term, see (1.15), is small as well. (As was pointed by A. Levitt, a result for small $\kappa = \kappa _{\mathrm{per}}$ would not work in Theorem 1.2 above as Assumption [A1] fails for it.)

The results given in Theorem 1.5 is taken from [13]. Papers [1, 10, 11, 13] use variational techniques and did not provide uniqueness results. A. Levitt [23] proved the screening of small defects for the TRHFE.

Approach As in [23], our starting point is Eq. (1.5) for the electrostatic potential $\phi$. We also use some important ideas from [8]. However, our approach to proving Theorem 1.2 is fairly novel. Rather that employing variations-based techniques, we use the Lyapunov–Schmidt reduction, which also allows us to estimate the remainders.

The starting equation of our analysis can be formulated as follows. Let $(\phi _{\mathrm{per}} (x), \mu _{\mathrm{per}})$ be the solution of (1.5), with $\kappa (x)=\kappa _{\mathrm{per}} (x)$, and let $\kappa _\delta$ be given in (1.23). Define $\psi$ by the equality

$$\begin{aligned} \phi = \phi _{\mathrm{per}} + \psi . \end{aligned}$$

Plugging this decomposition into (1.5), with $\kappa = \kappa _\delta$ and $\mu =\mu _{\mathrm{per}}$, and using that $h^{\phi }=h^{\phi _{\mathrm{per}}}-\psi$, we arrive at the equation for $\psi$:

$$\begin{aligned} -\Delta \psi = ({\kappa '}_\delta - {\text {den}}[g_{\phi _{\mathrm{per}}'}(\psi )]), \end{aligned}$$

(1.44)

where $\phi _{\mathrm{per}}':=\phi _{\mathrm{per}}+\mu _{\mathrm{per}}$, $g_{\phi _{\mathrm{per}}'}(\psi ):=f_T(h^{\phi _{\mathrm{per}}'}-\psi )-f_T(h^{\phi _{\mathrm{per}}'})$ and ${\kappa '}_\delta (y):=\delta ^{3}\kappa '(\delta y)$.

This is a nonlinear and nonlocal Poisson equation for $\psi$. We see that only $\phi _{\mathrm{per}}':=\phi _{\mathrm{per}}+\mu _{\mathrm{per}}$, but not $\kappa _{\mathrm{per}}$, enters Eq. (1.44) explicitly.

Though we deal with the simplest microscopic model—the reduced HF equation—our techniques are fairly robust and would work for the full-fledged DFT. Also, we favoured rough estimates to more precise but lengthier ones which produce better bounds on $\beta$ in (1.28), see Appendix D below.

The paper is organized as follows. After presenting preliminary material on charge density estimates and the Bloch–Floquet decomposition in Sect. 2, we prove Theorem 1.2 in Sects. 3–5. Section 3 contains the main steps of the proof of Theorem 1.2. Section 5 covers fairly straightforward technical estimates of the nonlinearity.

2 Densities and Bloch–Floquet Decomposition

2.1 Locally Trace Class Operators

Let $C_c\equiv C_c(\mathbb {R}^3)$ denote the space of compactly supported continuous functions on $\mathbb {R}^3$. An operator A on $L^2$ is said to be locally trace class if fA and Af are trace class for all $f \in C_c$. (For the proofs below, it suffices to require that fA is trace class.)

Let $\mathcal {L}$ be a Bravais lattice on $\mathbb {R}^3$ and $\Omega$ a fundamental domain of $\mathcal {L}$ as in Sect. 1.5. Denote |S| to be the volume of a measurable set $S \subset \mathbb {R}^3$ and note that $|\Omega |$ is independent of the choice of the fundamental cell $\Omega$. Let $T_s$ be the translation operator

$$\begin{aligned} T_s : f(x) \mapsto f(x-s). \end{aligned}$$

(2.1)

We say that a function $f: \mathbb {R}^3\rightarrow \mathbb {C}$ is $\mathcal {L}$-periodic if and only if it is invariant under the translations action of $T_s$ for all lattice elements $s \in \mathcal {L}$. We define the space

$$\begin{aligned} L^p_{\mathrm{per}}\equiv L^p_{\mathrm{per}}(\mathbb {R}^3) = \{ f \in L^p_{\mathrm{loc}}(\mathbb {R}^3) : f \text { is } \mathcal {L}\text {-periodic} \}, \end{aligned}$$

(2.2)

with the norm of $L^p(\Omega )$ for some $\Omega$. The norms for $L^p_{\mathrm{per}}$ and $L^p\equiv L^p(\mathbb {R}^3)$ are distinguished by the subindices $L^p_{\mathrm{per}}$ and $L^p$.

We say that a bounded operator A on $L^2$ is $\mathcal {L}$-periodic if and only if $[A, T_s] = 0$ for all $s \in \mathcal {L}$ where $T_s$ is the translation operator defined in (2.1).

Let $S^p$ be the standard p-Schatten space of bounded operators on $L^2$ with the p-Schatten norm

$$\begin{aligned} \Vert A\Vert _{S^p}^p :=&\mathrm {Tr}_{L^2}( (A^*A)^{p/2} ). \end{aligned}$$

(2.3)

Next, let $\chi _Q$ denote the characteristic function of a set $Q \subset \mathbb {R}^3$ and let $S^p_{\mathrm{per}}$ be the space of bounded, $\mathcal {L}$-periodic operators A on $L^2$ with $\Vert A\Vert _{S^p_{\mathrm{per}}} < \infty$ where

$$\begin{aligned} \Vert A\Vert _{S^p_{\mathrm{per}}}^p :=&\mathrm {Tr}_{\Omega }( (A^*A)^{p/2} ) := \frac{1}{|\Omega |} \mathrm {Tr}_{L^2}( \chi _\Omega (A^*A)^{p/2} \chi _\Omega ). \end{aligned}$$

(2.4)

We remark that the $S_{\mathrm{per}}^2$ norm does not depend on the choice of $\Omega$ since A is $\mathcal {L}$-periodic. We have the following estimates for the densities in terms of Schatten norms.

2.2 Densities

For a locally trace class operator A, we define its density ${\text {den}}[A]$ to be a regular countably additive complex Borel measure satisfying

$$\begin{aligned}&\int {\text {den}}(A)f= \mathrm {Tr}( fA), \end{aligned}$$

(2.5)

for every $f \in C_c$. If $\mathrm {Tr}(fA)$ is continuous in f in the $C_c$-topology, then the Riesz representation theorem shows that (2.5), for every $f \in C_c$, define ${\text {den}}[A]$ uniquely. In our case, we will frequently stipulate stronger regularity assumptions on A, implying that ${\text {den}}[A]$ is actually in a reasonable function space. (e.g. Lemma 2.1 below).

If an operator A has an (distributional) integral kernel, A(x, y), with the diagonal, A(x, x), being a regular countably additive complex Borel measure, then

$$\begin{aligned}&{\text {den}}(A)(x) = A(x, x). \end{aligned}$$

(2.6)

Finally, den is a linear map on locally trace class operators with the property that for any $f \in C_c$,

$$\begin{aligned}&{\text {den}}( fA) = f{\text {den}}(A). \end{aligned}$$

(2.7)

Lemma 2.1

Let A be a locally trace class operator on $L^2$ and $\epsilon > 0$. We have the following statements.

• (1) If $(1-\Delta )^{3/4 + \epsilon } A \in S^2$, resp. $S^2_{\mathrm{per}}$, then ${\text {den}}[A] \in L^2$, resp. $L^2_{\mathrm{per}}$. Moreover, respectively,

  $$\begin{aligned}&\Vert {\text {den}}[A]\Vert _{L^2} \lesssim \Vert (1-\Delta )^{3/4 + \epsilon } A\Vert _{S^2}, \end{aligned}$$

  (2.8)
  $$\begin{aligned}&\Vert {\text {den}}[A]\Vert _{L^2_{\mathrm{per}}} \lesssim |\Omega |^{1/2} \Vert (1-\Delta )^{3/4 + \epsilon } A\Vert _{S^2_{\mathrm{per}}} \end{aligned}$$

  (2.9)

• (2) If $(1-\Delta )^{1/4 + \epsilon } A \in S^{6/5}$, then ${\text {den}}[A] \in \dot{H}^{-1}$ (where $\dot{H}^s$ is defined in (1.24)). Moreover,

  $$\begin{aligned} \Vert {\text {den}}[A]\Vert _{\dot{H}^{-1}} \lesssim \Vert (1-\Delta )^{1/4 + \epsilon } A\Vert _{S^{6/5}} \end{aligned}$$

  (2.10)

Proof

We prove (2.9) and (2.10) only; (2.8) is similar and easier. We begin with (2.9). Since the operator $(1-\Delta )^{1/4+\epsilon } A$ is $\mathcal {L}$-periodic, its density, if it exists, is also $\mathcal {L}$-periodic. By the $L^2_{\mathrm{per}}$-$L^2_{\mathrm{per}}$ duality, relation (2.5), ${\text {den}}[A] \in L^2_{\mathrm{per}}$ and (2.9) holds if and only if

$$\begin{aligned} |\mathrm {Tr}_{\Omega }(f A)| \lesssim |\Omega |^{1/2} \Vert f\Vert _{L^2_{\mathrm{per}}}\Vert (1-\Delta )^{3/4 + \epsilon } A\Vert _{S^{\mathrm{per}}_2} \end{aligned}$$

(2.11)

for all $f \in L^2(\mathbb {R}^3)$ with support in $\Omega$, where we recall $\Vert f\Vert _{L^2_{\mathrm{per}}} = \Vert f\chi _\Omega \Vert _{L^2}$. Since the support of f is in $\Omega$, by the Hölder’s inequality for the trace-per-volume norm,

$$\begin{aligned} \frac{1}{|\Omega |}|\mathrm {Tr}_{\Omega }(f A)|&= \frac{1}{|\Omega |}| \mathrm {Tr}( \chi _\Omega fA \chi _\Omega )| \end{aligned}$$

(2.12)

$$\begin{aligned}&\lesssim \Vert A(1-\Delta )^{3/4+\epsilon }\Vert _{S^2_{\mathrm{per}}} \Vert (1-\Delta )^{-3/4-\epsilon }f\Vert _{S^2_{\mathrm{per}}}. \end{aligned}$$

(2.13)

By the Kato–Seiler–Simon inequality

$$\begin{aligned} \Vert f(x)g(-i\nabla ) \Vert _{S^p} \lesssim \Vert f\Vert _{L^p}\Vert g\Vert _{L^p} \end{aligned}$$

(2.14)

for $2 \le p < \infty$ (see [36]; one can also replace $S^p$ and $L^p$ by their periodic versions $S_{\mathrm{per}}^p$ and $L^p_{\mathrm{per}}$, respectively.), we obtain (2.11). Thus, (2.9) is proved.

Now we prove (2.10) as above. By the $\dot{H}^1$-$\dot{H}^{-1}$ duality, it suffices to show that

$$\begin{aligned} |\mathrm {Tr}(f A)| \lesssim \Vert f\Vert _{\dot{H}^1}\Vert (1-\Delta )^{1/4 + \epsilon } A\Vert _{S^{6/5}} \end{aligned}$$

(2.15)

for all $f \in \dot{H}^1 \cap C_c$ and for $\epsilon > 0$. So, we estimate $|\mathrm {Tr}(f A)|$. By the non-abelian Hölder inequality with $1 = \frac{1}{6} + \frac{1}{6/5}$ ( [36]),

$$\begin{aligned} |\mathrm {Tr}(f A)| \lesssim \Vert f(1-\Delta )^{-1/4 - \epsilon }\Vert _{S^6} \Vert (1-\Delta )^{1/4 + \epsilon } A\Vert _{S^{6/5}} . \end{aligned}$$

(2.16)

The Kato–Seiler–Simon inequality (2.14) shows

$$\begin{aligned} |\mathrm {Tr}(f A)| \lesssim \Vert f\Vert _{L^6} \Vert (1-\Delta )^{1/4 + \epsilon } A\Vert _{S^{6/5}}. \end{aligned}$$

(2.17)

Now, applying the Gagliardo–Nirenberg–Sobolev inequality (for $d=3$; see [24])

$$\begin{aligned} \Vert f \Vert _{L^6} \lesssim \Vert \nabla f\Vert _{L^2} \end{aligned}$$

(2.18)

to $\Vert f\Vert _{L^6}$ in (2.17), we obtain (2.15). The proof of Lemma 2.1 is completed by the $\dot{H}^1$-$\dot{H}^{-1}$ duality and the fact that $\dot{H}^1 \cap C_c$ is dense in $\dot{H}^1$. $\square$

2.3 Bloch–Floquet Decomposition

Let $\mathcal {L}^*$ denote the lattice reciprocal to $\mathcal {L}$, with the reciprocity relation between bases for $\mathcal {L}$ and $\mathcal {L}^*$ given by $\omega _i\cdot \omega _j^*=2\pi \delta _{ij}$. Define the (fiber integral) space

$$\begin{aligned} \mathcal {H}_{\mathcal {L}}^\oplus&= \{ f \in L^2_{\mathrm{loc}} (\mathbb {R}^3_k \times \mathbb {R}^3_x) : T_s^x f= f \end{aligned}$$

(2.19)

$$\begin{aligned} \text { and } T_r^k f&= e^{- i r \cdot x} f,\ \,\forall s \in \mathcal {L},\ \forall r \in \mathcal {L}^* \}, \end{aligned}$$

(2.20)

where $T_s^k$ is the translation in the k-variable by s and $T_r^x$ is the translation in the x-variable by r (see (2.1)). We write $f = f_k(x) \in \mathcal {H}_{\mathcal {L}}^\oplus$ as

$$\begin{aligned} f=\int _{\mathbb {R}^3/\mathcal {L}^*}^\oplus \, f_k \, d{\hat{k}} = \int _{\Omega ^*}^\oplus \, f_k \, d{\hat{k}}, \end{aligned}$$

(2.21)

for some choice of a fundamental cell $\Omega ^*$ of the reciprocal lattice $\mathcal {L}^*$ and $d{\hat{k}} := |\Omega ^*|^{-1} dk$.

We use the Bloch–Floquet decomposition $U_{\mathrm{BF}}$ mapping from $L^2(\mathbb {R}^3)$ into $\mathcal {H}^\oplus _{\mathcal {L}}$ as

$$\begin{aligned} U_{\mathrm{BF}}f&:= \int ^\oplus _{\Omega ^*} d{\hat{k}} f_k \, , \end{aligned}$$

(2.22)

$$\begin{aligned} f_k(x)&:= \sum _{t \in \mathcal {L}} e^{- ik(x+t)} f(x+t) \end{aligned}$$

(2.23)

and the inverse Bloch–Floquet transform

$$\begin{aligned}&U_{\mathrm{BF}}^{-1}\left( \int ^\oplus _{\Omega ^*} d{\hat{k}} f_k \right) (x) := \int _{\Omega ^*} d{\hat{k}} \, e^{ ikx} f_k(x), \, \forall x\in \mathbb {R}^3. \end{aligned}$$

(2.24)

Lemma 2.2

We have, for any $f \in L^2$,

$$\begin{aligned} \int _{\Omega } f_k(x) dx = \hat{f}(k) \end{aligned}$$

(2.25)

Proof

By (2.23) and a change of variable, we see that

$$\begin{aligned} \int f_k(x) dx&:= \int _{\Omega } \sum _{t \in \mathcal {L}} e^{- ik(x+t)} f(x+t) d{x} \end{aligned}$$

(2.26)

$$\begin{aligned}&= \sum _{t \in \mathcal {L}} \int _{t+ \Omega } e^{- ikx} f(x) d{x} \end{aligned}$$

(2.27)

$$\begin{aligned}&= \int e^{- ikx} f(x) dx. \end{aligned}$$

(2.28)

Equation (2.25) follows from the definition of the Fourier transform. $\square$

Let $\langle f\rangle _{S} = |S|^{-1} \int _S f(x) dx$, the average of f on a set S, and $\chi _S$ be the indicator (characteristic) function of S.

Lemma 2.3

Let $f \in L^2$ and $f_k$ be its k-th fiber $\mathcal {L}$-Bloch–Floquet decomposition. Then for any $S \subset \Omega ^*$,

$$\begin{aligned} \chi _S(-i\nabla )f = U_{\mathrm{BF}}^{-1}\int ^{\oplus }_{S} d{\hat{k}} \, \langle f_k \rangle _{\Omega }. \end{aligned}$$

(2.29)

Proof

Let $f \in L^2$ with the k-th fiber $f_k$. Then Lemma 2.2 shows that

$$\begin{aligned} \langle f_k \rangle _{\Omega } = |\Omega |^{-1} \hat{f}(k). \end{aligned}$$

(2.30)

Using the definition of the inverse Bloch transform in (2.24) and (2.30), we see that

$$\begin{aligned} U_{\mathrm{BF}}^{-1} \left( \int _{S}^\oplus d{\hat{k}} \, \langle f_k \rangle _{\Omega } \right)&= \int _{\Omega ^*} d{\hat{k}} \, e^{ ikx} \langle f_k \rangle _{\Omega _\delta } \nonumber \\&= \int _{S} d{\hat{k}} \, |\Omega |^{-1} e^{ ikx} {\hat{f}} (k) \end{aligned}$$

(2.31)

Since $d{\hat{k}} = |\Omega ^*|^{-1}dk = |\Omega | dk$, the last equation yields

$$\begin{aligned} U_{\mathrm{BF}}^{-1} \int _{S}^\oplus d{\hat{k}} \, \langle f_k \rangle _{\Omega } =&\int _S d k \, e^{ ikx}\hat{f}(k) = \chi _S(-i\nabla ) f, \end{aligned}$$

(2.32)

which gives (2.29). $\square$

Let $P_r= \chi _{B(r)}(-i\nabla )$ where B(r) is the ball of radius r centered at the origin (see (3.37)). Lemmas 2.2 and 2.3 imply

Corollary 2.4

Let $f \in L^2$ and $B(r)\subset \Omega ^*$, then

$$\begin{aligned} (P_rf)_k = |\Omega |^{-1} {\hat{f}}(k)\chi _{B(r)}(k). \end{aligned}$$

(2.33)

Any $\mathcal {L}$-periodic operator A has a Bloch–Floquet decomposition [34] in the sense that

$$\begin{aligned} A = U_{\mathrm{BF}}^{-1} \int ^\oplus _{\Omega ^*} d{\hat{k}} A_k U_{\mathrm{BF}}, \end{aligned}$$

(2.34)

where $A_k$ are operators (called k-fibers of A) on $L^2_{\mathrm{per}}$ and the operator $\int ^\oplus _{\Omega ^*} d{\hat{k}} \, A_k$ acts on $\int ^\oplus _{\Omega ^*} d{\hat{k}} f_k \in \mathcal {H}_{\mathcal {L}}^\oplus$ as

$$\begin{aligned} \int ^\oplus _{\Omega ^*} d{\hat{k}} A_k \cdot \int ^\oplus _{\Omega ^*} d{\hat{k}} f_k = \int ^\oplus _{\Omega ^*} d{\hat{k}} A_kf_k. \end{aligned}$$

(2.35)

Definitions (2.34) and (2.35) implies the following relations for any $\mathcal {L}$-periodic operators A and B

$$\begin{aligned}&(A f)_k= A_k f_k, \end{aligned}$$

(2.36)

$$\begin{aligned}&(A B)_k= A_k B_k, \end{aligned}$$

(2.37)

$$\begin{aligned}&\Vert A \Vert = \sup _{k\in \Omega ^*}\Vert A_k \Vert . \end{aligned}$$

(2.38)

Furthermore, we have

Lemma 2.5

Let A be an $\mathcal {L}$-periodic operator and $A_k$, its k-fibers in its Bloch–Floquet decomposition. Then

$$\begin{aligned} A_k = e^{- ixk}A_{0} e^{ ixk}. \end{aligned}$$

Proof

We compute $(Af)_k$. Let $T_s$ denote the translation operator (2.1). Let $A_0$ denote the 0-th fiber of A in its Bloch–Floquet decomposition. By (2.23) and the periodicity of A,

$$\begin{aligned} (Af)_k&= \sum _{t\in \mathcal {L}} e^{- ik(x+t)} T_{-t} Af = \sum _{t\in \mathcal {L}} e^{- ikx} A e^{- i kt} T_{-t} f \end{aligned}$$

(2.39)

$$\begin{aligned}&= e^{- ikx} A_0 e^{ ikx} \sum _{t\in \mathcal {L}} e^{- i k(x+t)} T_{-t} f \end{aligned}$$

(2.40)

$$\begin{aligned}&= e^{- ikx} A_0 e^{ ikx} f_k. \end{aligned}$$

(2.41)

$\square$

Now, we have the following result.

Lemma 2.6

Let A be an $\mathcal {L}$-periodic operator and $A_k$, its k-fibers in its Bloch–Floquet decomposition and let r be such that $B(r)\subset \Omega ^*$. Then

$$\begin{aligned} P_rA P_r= b(-i \nabla ) P_r\end{aligned}$$

(2.42)

where $b (k) = \langle A_k \mathbf {1}\rangle _{\Omega }$, $1 \in L^2_{\mathrm{per}}(\mathbb {R}^3)$ is the constant function 1.

Proof

Let $f_k$ be the k-th fiber of the Bloch–Floquet function f. We apply Lemma 2.3 with $S = B(r)$ and $f = A P_r\varphi$ (so that $\chi _S(-i\nabla ) = P_r$) to obtain

$$\begin{aligned} P_rA P_r\varphi&= U_{\mathrm{BF}}^{-1} \int ^{\oplus }_{\Omega ^*} d{\hat{k}} \, \langle (A P_r\varphi )_k \rangle _{\Omega }. \end{aligned}$$

(2.43)

By Corollary 2.4 and Eq. (2.43), we find

$$\begin{aligned} P_rA P_r\varphi&= |\Omega |^{-1} U_{\mathrm{BF}}^{-1} \int ^{\oplus }_{B(r)} d{\hat{k}} \, \langle A_k 1 \rangle _{\Omega } {\hat{\varphi }}(k), \end{aligned}$$

(2.44)

where $1 \in L^2_{\mathrm{per}, \delta }$ is the constant function equal to 1. Using the definition (2.24) of the inverse Bloch–Floquet transform and that $d {\hat{k}} = |\Omega |^{-1} dk$, we deduce (2.48). $\square$

2.4 Passing to the Macroscopic Variables

Define the microscopic lattice $\mathcal {L}_\delta :=\delta \mathcal {L}$ and let $\mathcal {L}_\delta ^*$ be its reciprocal lattice. Define the rescaling operator

$$\begin{aligned} U_{\delta }: f(x) \mapsto \delta ^{-3/2} f(\delta ^{-1} x) \end{aligned}$$

(2.45)

mapping from the microscopic to the macroscopic scale. A change of variable in (2.5) gives the following

Lemma 2.7

For any operator A on $L^2$, we have

$$\begin{aligned} \delta ^{-3/2}U_{\delta }{\text {den}}[A] = {\text {den}}[U_{\delta }AU_{\delta }^*]. \end{aligned}$$

(2.46)

Finally, note that

$$\begin{aligned} A\ \text { is } \mathcal {L}\text {-periodic iff } U_{\delta }A U_{\delta }^* \text { be } \mathcal {L}_\delta \text {-periodic}. \end{aligned}$$

(2.47)

Lemma 2.6 implies

Lemma 2.8

Let A be an $\mathcal {L}$-periodic operator and $A_k$, its k-fibers in its Bloch–Floquet decomposition and let r be such that $B(\delta r)\subset \Omega ^*$. Then

$$\begin{aligned} P_rU_{\delta }A U_{\delta }^* P_r= b(-i\delta \nabla ) P_r\end{aligned}$$

(2.48)

where $b (k) = \langle A_k \mathbf {1}\rangle _{\Omega }$, $1 \in L^2_{\mathrm{per}}$ is the constant function 1.

Proof

By $U_{\delta }^* P_rU_{\delta }= P_{\delta r}$ and Lemma 2.6, we have

$$\begin{aligned} P_rU_{\delta }A U_{\delta }^* P_r=U_{\delta }P_{\delta r} A P_{\delta r}U_{\delta }^* = U_{\delta }b(-i \nabla ) P_{\delta r}U_{\delta }^*. \end{aligned}$$

Relations $U_{\delta }P_{\delta r} U_{\delta }^* = P_r$ and $U_{\delta }b(-i \nabla ) U_{\delta }^* = b(-i\delta \nabla )$ yield (2.48). $\square$

3 Dielectric Response: Proof of Theorem 1.2

In this section, we prove Theorem 1.2 modulo several technical (though important) statements proved in Sects. 4 and 5.

3.1 Linearized Map

Our starting point is Eq. (1.5), which we reproduce here

$$\begin{aligned} -\Delta \phi = (\kappa - {\text {den}}[f_T(h^{\phi }-\mu )]), \end{aligned}$$

(3.1)

where, recall, $f_{T}(\lambda )$ is given in (1.2) and, recall,

$$\begin{aligned} h^{\phi } := -\Delta - \phi . \end{aligned}$$

(3.2)

We consider (3.1) on the function space $\phi \in H_{\mathrm{per}}^2 + \dot{H}^1$. For such $\phi$’s, the operator $h^{\phi }$ is self-adjoint and bounded below so that functions of $h^{\phi }$ above are well-defined by the spectral theory.

Our first step is to investigate the linearization of the map on the r.h.s. of (3.1)

$$\begin{aligned}&M:= d_\phi {\text {den}}[f_T(h^{\phi } - \mu )]\big |_{\phi =\phi _{\mathrm{per}}}. \end{aligned}$$

(3.3)

To derive basic properties of M, we find an explicit formula for it. Recalling the relation $f_T(\lambda ):=f_{FD}(\lambda /T)$, see (1.2), and assuming that $\phi$ is close to $\phi _{\mathrm{per}}$, we write $f_T(h^{\phi } - \mu )$ using the Cauchy-integral formula

$$\begin{aligned} f_T(h^{\phi } - \mu ) = \frac{1}{2\pi i} \int _\Gamma dz f_T(z-\mu ) (z-h^{\phi })^{-1} \end{aligned}$$

(3.4)

where $\Gamma$ is a positively oriented contour around the spectrum of $h^{\phi }$ not containing the poles of $f_T$ which are located at $\mu + i\pi (2k+1) T$, $k \in \mathbb {Z}$ (see Fig. 2 below), in which $\epsilon$ satisfies

$$\begin{aligned} \epsilon< T \pi \ \text { and }\ -1 < \cos (\mu \epsilon ). \end{aligned}$$

(3.5)

Here we use that $h^\phi$ is bounded from below and, due to the definition $f_T(\lambda ) = \frac{1}{e^{\lambda /T}+1}$ (see (1.2)) and the relation $|\mathrm {Im}z|\le \pi /4 T$,

$$\begin{aligned}&|f_T(z-\mu )| \lesssim \min (1, e^{-(\mathrm {Re}z-\mu )/T})) \end{aligned}$$

(3.6)

assuring the convergence of the integral. (Note that we do not use that $h^\phi$ has a gap and that $\mu$ is in the gap.)

Fig. 1

The contour $\Gamma$. It depends on $\epsilon$ satisfying (3.5)

To simplify the expressions below, we will introduce the following notation

$$\begin{aligned} \oint :=&\frac{1}{2\pi i} \int _\Gamma dz f_T(z-\mu ) \end{aligned}$$

(3.7)

where $\Gamma$ is the contour given in Fig. 1, with the positive orientation.

Recall the notation for the $\mathcal {L}$-periodic Hamiltonian and introduce one for the $\mathcal {L}$-periodic resolvent:

$$\begin{aligned} h_{\mathrm{per}}:= h^{\phi _{\mathrm{per}}}= -\Delta -\phi ^{\mathrm{per}}, \quad r_{\mathrm{per}}(z) = (z-h_{\mathrm{per}})^{-1}. \end{aligned}$$

(3.8)

By Theorem 1.5, the electrostatic potential, $\phi _{\mathrm{per}}(y)$ associated with the solution $\rho _{\mathrm{per}}(y)$ (c.f. (1.9)) satisfies

$$\begin{aligned} \phi _{\mathrm{per}} \in H^2_{\mathrm{per}}. \end{aligned}$$

(3.9)

Hence the operator $h_{\mathrm{per}}$ is self-adjoint and the operator functions above are well-defined. Moreover, under Assumption [A1],

$$\begin{aligned} \sup _{z\in \Gamma } \Vert (z-h_{\mathrm{per}})^{-1}\Vert _\infty = O(1). \end{aligned}$$

(3.10)

Finally, for any operator h, we denote $h^L: \alpha \rightarrow h\alpha$ and $h^R: \alpha \rightarrow \alpha h$.

The next proposition gives an explicit form for M and states its properties (also see [7]).

Proposition 3.1

Let Assumption [A1] hold. Then

• (1) The operator M has the following explicit representation

  $$\begin{aligned} M f&= - {\text {den}}\big [ \oint r_{\mathrm{per}}(z) f r_{\mathrm{per}}(z) \big ] \end{aligned}$$

  (3.11)
  $$\begin{aligned}&=- \frac{1}{2}{\text {den}}\big [ \frac{\tanh (\frac{1}{2 T}(h_{\mathrm{per}}^L-\mu )) - \tanh (\frac{1}{2 T}(h_{\mathrm{per}}^R-\mu ))}{h_{\mathrm{per}}^L- h_{\mathrm{per}}^R}f \big ], \end{aligned}$$

  (3.12)

  where $f \in L^2$ on the right hand side is considered as a multiplication operator.

• (2) The operator M is bounded, self-adjoint, positive on $L^2$ and $\mathcal {L}$-periodic (c.f. Sect. 2.2) and satisfies

  $$\begin{aligned} \Vert M\Vert \lesssim 1. \end{aligned}$$

  (3.13)

Proof of Proposition 3.1

In this proof, we omit the subscript “per” in $h_{\mathrm{per}}$ and $r_{\mathrm{per}}(z)$. We begin with item (1). Equation (3.11) follows from definition (3.3), the Cauchy formula (3.4) and a simple differentiation of the resolvent.

Now, we use (3.11) to derive (3.12). By the definition of $h^L$ and $h^R$ and the second resolvent identity, we have, for any operator $\alpha$,

$$\begin{aligned}&(z-h)^{-1} \alpha (z-h)^{-1} = (z-h^L)^{-1} (z-h^R)^{-1} \alpha \nonumber \\&\quad =(h^L-h^R)^{-1} ((z-h^L)^{-1} - (z-h^R)^{-1}) \alpha . \end{aligned}$$

(3.14)

Using the Cauchy integral formula and the definition (3.7) and the choice of the contour $\Gamma$ i(see Fig. 1), we observe that

$$\begin{aligned}&\oint (z-h)^{-1} \alpha (z-h)^{-1}= (h^L-h^R)^{-1} \nonumber \\&\qquad \times \frac{1}{2\pi i} \int _\Gamma dz f_T(z-\mu ) ((z-h^L)^{-1} - (z-h^R)^{-1}) \alpha \nonumber \\&\quad = (h^L-h^R)^{-1} (f_T(h^L-\mu ) - f_T(h^R-\mu )) \alpha . \end{aligned}$$

(3.15)

Now, by definition (1.2), $f_T(\lambda ):= \frac{1}{e^{\lambda /T}+1}$ and therefore $f_T(\lambda )=\frac{1}{2}(1+\tanh (\lambda /2T))$. This relation, together with (3.15), gives

$$\begin{aligned}&\oint (z-h)^{-1} \alpha (z-h)^{-1} \nonumber \\&\quad = \frac{1}{2}\frac{\tanh (\frac{1}{2 T}(h^L-\mu )) - \tanh (\frac{1}{2 T}(h^R-\mu ))}{h^L- h^R} \alpha . \end{aligned}$$

(3.16)

This, together with (3.11), gives (3.12). Item (1) is now proved.

Now we prove item (2). Since $h = h_{\mathrm{per}}$ is self-adjoint and bounded below, we can pick $c>0$ sufficiently large, s.t. $h\ge - c+1$. Then, in particular, $h+c$ is invertible and, for each function $f \in L^2(\mathbb {R}^3)$, we define the operator

$$\begin{aligned} \alpha _f := (c+h)^{-1/2} f (c+h)^{-1/2}. \end{aligned}$$

(3.17)

The Kato–Seiler–Simon inequality (2.14) shows that $\alpha _f$ is Hilbert–Schmidt and

$$\begin{aligned} \Vert \alpha _f\Vert _{S^2} \lesssim \Vert f\Vert _{L^2} \end{aligned}$$

(3.18)

(the $S^2$ norm is given in (2.3)). Using (3.11), together with (2.5), we write $\langle f, Mg \rangle$

$= -\oint \mathrm {Tr}( {\bar{f}} r (z) g r(z))$, which can be transformed to

$$\begin{aligned} \langle f, Mg \rangle =&- \oint \mathrm {Tr}( \alpha _{f}^* (c+h) r(z) \alpha _{g} r(z) (c+h)) \, . \end{aligned}$$

(3.19)

Moreover, by (3.16), we have that

$$\begin{aligned}&\langle f, M g \rangle = \mathrm {Tr}\left( \alpha _{f}^* G(h^L, h^R) \alpha _{g} \right) , \end{aligned}$$

(3.20)

$$\begin{aligned}&G(x,y) := -\frac{1}{2} \frac{\tanh (\frac{1}{2 T}(x-\mu ))-\tanh (\frac{1}{2 T}(y-\mu ))}{(x+c)^{-1}-(y+c)^{-1}}. \end{aligned}$$

(3.21)

Since the function $G : \mathbb {R}^2 \rightarrow \mathbb {R}$ is bounded on the set $x,y \ge -c+1$, we see that M is bounded due to (3.18) and (3.20).

Moreover, we can also see from expressions (3.20) - (3.21) that $M$ is symmetric since G is real and $h^L$ and $h^R$ are self-adjoint in the space $S^2$. Since M is bounded, it is self-adjoint. Since the function G in (3.21) is positive for $x,y \ge -c+1$, Eq. (3.20) and spectral theorem on $S^2$ show that $\langle f, Mf \rangle =\mathrm {Tr}\left( \alpha _{f}^* G(h^L, h^R) \alpha _{f} \right) > 0$ for any nonzero $f \in L^2(\mathbb {R}^3)$. This shows that M is positive.

Finally, formula (3.11) and the fact $h = h_{\mathrm{per}}$ and $r = r_{\mathrm{per}}(z)$ are $\mathcal {L}$-periodic show that M is $\mathcal {L}$-periodic.

To prove bound (3.13), we use (3.11) and (2.8) to find

$$\begin{aligned} \Vert M f\Vert _{L^2}&\lesssim \Vert (1-\Delta ) \oint r (z) f r (z)\Vert _{S^2} \end{aligned}$$

(3.22)

$$\begin{aligned}&\lesssim \big |\oint \big |\Vert (1-\Delta ) r (z)\Vert ^2 \Vert f (1-\Delta )^{-1}\Vert _{S^2} \, . \end{aligned}$$

(3.23)

Now, writing $-\Delta =h-z+\phi _{\mathrm{per}}+z,$ for $z\in \Gamma$, and using the uniform boundedness of $\Vert \phi _{\mathrm{per}}\Vert _{H^2}$ which follows from Assumption [A1], we derive the estimate

$$\begin{aligned} \Vert (1-\Delta ) r(z)\Vert \lesssim \&1, \end{aligned}$$

(3.24)

which, together with (3.23) and the Kato-Seiler-Simon inequality (2.14), gives bound (3.13). The proof of Proposition 3.1 is now complete. $\square$

3.2 Scaling and Splitting

This step is to pass from the microscopic coordinate y to the macroscopic one, $x= \delta y$ passing to the macroscopic quantities (with superscripts $\delta$) which are related the microscopic quantities (with subscripts $\delta$) as

$$\begin{aligned}&\kappa ^\delta = \delta ^{-3/2}U_{\delta }\kappa _\delta ,\ \phi ^\delta (x) = \delta ^{1/2} (U_{\delta }\phi _\delta )(x)= \delta ^{-1} \phi _\delta (\delta ^{-1}x ), \end{aligned}$$

(3.25)

$$\begin{aligned}&\kappa _{\mathrm{per}}^\delta (x) := \delta ^{-3} \kappa _{\mathrm{per}}(\delta ^{-1}x ) = (\delta ^{-3/2}U_{\delta }\kappa _{\mathrm{per}})(x), \end{aligned}$$

(3.26)

where $U_{\delta }: f(x) \mapsto \delta ^{-3/2} f(\delta ^{-1} x)$, the $L^2(\mathbb {R}^3)$-unitary scaling map, see (2.45) (note that the $L^1$-norm, hence total charge, is preserved under this scaling). Let

$$\begin{aligned} \kappa ^\delta (x) = \kappa _{\mathrm{per}}^\delta (x) + \kappa '(x) \end{aligned}$$

(3.27)

be the macroscopic perturbed background potential. Accordingly, we rescale equation (3.1) by applying $\delta ^{-3/2}U_{\delta }$ to it. Using Lemma 2.7 and relations $U_{\delta }f_{\mathrm{T}}(h^{\phi }-\mu )U_{\delta }^*=f_{\mathrm{T}}(U_{\delta }h^{\phi }U_{\delta }^*-\mu )$ and

$$\begin{aligned} U_{\delta }h^{\phi }U_{\delta }^* = -\delta ^2 \Delta - \delta \phi ^\delta , \end{aligned}$$

we arrive at the rescaled electrostatic potential equation

$$\begin{aligned}&- \Delta \phi ^\delta = \kappa ^\delta - F_\delta (\phi ^\delta ), \end{aligned}$$

(3.28)

$$\begin{aligned}&F_\delta (\phi ) = {\text {den}}[ f_T(-\delta ^2 \Delta - \delta \phi - \mu )]. \end{aligned}$$

(3.29)

We will consider (3.28) on the space $H_{\mathrm{per}}^2 +\dot{H}^1$.

Let $\phi _{\mathrm{per}}^\delta = \delta ^{1/2} U_{\delta }\phi _{\mathrm{per}}$, where $\phi _{\mathrm{per}}$ is the periodic potential associated to the periodic solution $(\rho _{\mathrm{per}}, \mu _{\mathrm{per}}$) of (1.1) with periodic background charge $\kappa _{\mathrm{per}}$ given in Theorem 1.5. We split the solution $\phi ^\delta$ into the big part $\phi _{\mathrm{per}}^\delta$ and the fluctuation

$$\begin{aligned} \varphi \equiv \varphi ^\delta := \phi ^\delta - \phi _{\mathrm{per}}^\delta . \end{aligned}$$

(3.30)

We rewrite Eq. (3.28) by expanding the r.h.s. around $\phi _{\mathrm{per}}^\delta$ to obtain

$$\begin{aligned} K_\delta \varphi = {\kappa '}+ N_\delta (\varphi ) \end{aligned}$$

(3.31)

where $N_\delta$ is defined by this expression and

$$\begin{aligned} K_\delta =&-\Delta + M_\delta \, ,\ \text { with }\ M_\delta = d_\phi F_\delta (\phi _{\mathrm{per}}^\delta ). \end{aligned}$$

(3.32)

Note that the inputs into this equation are $\phi _{\mathrm{per}},\ \mu =\mu _{\mathrm{per}}$ and ${\kappa '}$ (cf. (1.44)).

As was mentioned in the introduction, we prove Theorem 1.2 by decomposing $\varphi$ in (3.30) in small and large momentum parts (c.f. [8]). We use rough estimates for high momenta while we expand in $\delta$ and use a perturbation argument for low momenta.

We begin with a discussion of the linearized map, $K_\delta$. Since we rescaled equation (1.1) by applying $\delta ^{-3/2}U_{\delta }$ to it and rescaled the microscopic potentials via (3.25), it follows that

$$\begin{aligned} F_\delta = \delta ^{-3/2} U_{\delta }\circ F \circ (\delta ^{-1/2}U_{\delta }^*) \end{aligned}$$

(3.33)

where $F = F_{\delta = 1}$. Thus, by the definition of $M_\delta$ in (3.32) and the fact it is linear, it can be written as

$$\begin{aligned} M_\delta = \delta ^{-2}U_{\delta }MU_{\delta }^* , \end{aligned}$$

(3.34)

where $M:=M_{\delta = 1}$ and is given by (3.3).

Recall that an operator A on $L^2(\mathbb {R}^3)$ is said to be $\mathcal {L}$-periodic if and only if it commutes with the translations $T_s$ (see (2.1)) by all lattice elements $s \in \mathcal {L}$. As an immediate consequence of Proposition 3.1, representation (3.11), and the rescaling (3.34), we have the following result

Proposition 3.2

Let Assumption [A1] hold. Then $M_\delta$ is $\mathcal {L}_\delta$-periodic, positive (so that $K_\delta = -\Delta + M_\delta > -\Delta$), bounded on $L^2$ with an $O(\delta ^{-2})$ bound, and has the following representation

$$\begin{aligned} M_\delta \varphi = - \delta {\text {den}}\left[ \oint r^\delta _{\mathrm{per}}(z)\varphi r^\delta _{\mathrm{per}}(z)\right] \, , \end{aligned}$$

(3.35)

where the resolvent operator $r^\delta _{\mathrm{per}}(z)$ acting on $L^2(\mathbb {R}^3)$ is given by

$$\begin{aligned}&r^\delta _{\mathrm{per}}(z) = (z-h^\delta _{\mathrm{per}})^{-1},\ \quad h^\delta _{\mathrm{per}}= -\delta ^2 \Delta - \delta \phi _{\mathrm{per}}^\delta . \end{aligned}$$

(3.36)

3.3 Lyapunov–Schmidt Decomposition

To separate small and large momenta, we now perform a Lyapunov–Schmidt reduction.

Let $\chi _{Q}$ be the characteristic function of a set $Q \subset \mathbb {R}^3$. Let $\Omega _\delta ^*$ denote the fundamental domain of $\mathcal {L}_\delta ^*$ as in Sect. 2.4. We recall the definition of the orthogonal projection onto low momenta (as [8])

$$\begin{aligned} P_r= \chi _{B(r)}(-i\nabla ) \, , \end{aligned}$$

(3.37)

where $B(r)$ is the ball of radius r centred at the origin. With m given in (1.33) and estimated in (1.34), we choose r such that $B(r) \subset \Omega _\delta ^*$ and

$$\begin{aligned} a:=\delta r = O(1) \text { small, but }\ a^4 \gg m, \end{aligned}$$

(3.38)

is independent of $\delta$ and T (or m) and is fixed. Below, we use the convention that $\lesssim$ is independent of r, $\delta$ and T. Let

$$\begin{aligned} {\bar{P_r}} = 1- P_r \end{aligned}$$

(3.39)

be the orthogonal projection onto the large momenta. We decompose

$$\begin{aligned} \varphi = \varphi _s + \varphi _l, \end{aligned}$$

(3.40)

where $\varphi _s = P_r\varphi$ and $\varphi _l = {\bar{P_r}}\varphi$. Here s stands for small momentum and l stands for large momenta. We split (3.31) as

$$\begin{aligned}&P_rK_\delta (\varphi _s + \varphi _l) = P_r{\kappa '}+ P_rN_\delta (\varphi ), \end{aligned}$$

(3.41)

$$\begin{aligned}&{\bar{P_r}} K_\delta (\varphi _s + \varphi _l) = {\bar{P_r}} {\kappa '}+ {\bar{P_r}} N_\delta (\varphi ) \, . \end{aligned}$$

(3.42)

We solve (3.42) for $\varphi _l$ in the ball

$$\begin{aligned} B_{l, \delta } :=&\{\varphi \in {\bar{P_r}} H^1 : \Vert \varphi \Vert _{\dot{H}^1} \le c_l\}, \end{aligned}$$

(3.43)

while keeping $\varphi _s$ fixed in the (deformed) ball

$$\begin{aligned}&B_{s,\delta } := \{ \varphi \in P_rH^1 : \Vert \varphi \Vert _{\delta } \le c_s \}, \end{aligned}$$

(3.44)

with the norm $\Vert \varphi \Vert _{\delta }$ given by

$$\begin{aligned} \Vert \varphi \Vert _{\delta }^2 :=\sum _0^1\zeta ^{2(i-1)} \Vert \nabla ^i\varphi \Vert _{L^2}^2 ,\ \zeta :=\delta m^{-1/2} . \end{aligned}$$

(3.45)

The constants $c_s$ and $c_l$ above (should not be confused with the estimating function$c_T$which appeared in Theorem1.2) are chosen to satisfy the conditions

$$\begin{aligned} \zeta \ll c_s \ll c_l \ll \theta ^{- 3/2}\zeta , \end{aligned}$$

(3.46)

where $\theta := m^{- 8/9} \delta$ and, recall, $\zeta :=\delta m^{-1/2}$.

The latter condition can be satisfied, provided

$$\begin{aligned} \theta := m^{-8/9} \delta \ll 1. \end{aligned}$$

(3.47)

Due to estimate (1.34), this is equivalent to condition (1.28).

We see that, while our model is parametrized by $\delta$ and $\beta$ satisfying (1.28), our method is determined by the parameters a, $c_s$ and $c_l$, satisfying (3.38) and (3.46).

The subleading term, $\psi$, in (1.29) just fits into $B_{s,\delta }$: $\Vert \psi \Vert _{\delta }\sim \zeta \ll c_s$. Finally, we note that since $\nabla ^{-1} {\bar{P_r}}\le r^{-1} {\bar{P_r}}, \nabla ^{-1}:=\nabla \Delta ^{-1}, r=a/\delta$, we have

$$\begin{aligned} \Vert \varphi \Vert _{L^2} \lesssim m^{ 1/2}\zeta \Vert \varphi \Vert _{\dot{H}^1}, \quad \Vert \varphi \Vert _{\delta } \lesssim \Vert \varphi \Vert _{\dot{H}^1} ,\quad \forall \varphi \in {\text {Ran}}{\bar{P_r}}. \end{aligned}$$

(3.48)

Equation (3.48) shows that, if $m^{ 1/2}\zeta =\delta \ll c_s/c_l$, then, in the $L^2$-norm, $B_{l,\delta }$ is much smaller that $B_{s,\delta }$.

In the proofs below, we will use the convention $\Vert \cdot \Vert _{\dot{H}^{0}}\equiv \Vert \cdot \Vert _{L^2}$ and the estimates of the nonlinearity $N_\delta$ (defined implicitly through (3.31)) proved in Proposition 5.2 in Sect. 5 below, under Assumption [A1]:

$$\begin{aligned}&\Vert N_\delta (\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2} \nonumber \\ {}&\quad \lesssim m^{-\frac{1}{3}} \delta ^{-1/2} (\Vert \varphi _1\Vert _{{ \delta }} + \Vert \varphi _2\Vert _{{ \delta }}) \Vert \varphi _1 - \varphi _2\Vert _{{ \delta }}.\ \end{aligned}$$

(3.49)

Proposition 3.3

Let Assumptions [A1]–[A3] hold. Assume $\varphi _s \in B_{s,\delta }$ and that (3.38) holds. Then Eq. (3.42) on $B_{l,\delta }$ has a unique solution $\varphi _l = \varphi _l(\varphi _s)\in B_{l,\delta }$.

Proof of Proposition 3.3

We use that, by Proposition 3.2, ${\bar{K}}_\delta :={\bar{P_r}} K_\delta {\bar{P_r}}$ is invertible on the range of ${\bar{P_r}}$ (see (3.39)) to convert (3.42) into a fixed point problem

$$\begin{aligned}&\varphi _l = \Phi _l(\varphi _l) =\Phi _l' + \Phi _l''(\varphi _l), \end{aligned}$$

(3.50)

where

$$\begin{aligned}&\Phi _l':= {\bar{K}}_\delta ^{-1} (-M_\delta \varphi _s +{\bar{P_r}} {\kappa '}), \end{aligned}$$

(3.51)

$$\begin{aligned}&\Phi _l''(\varphi _l):= {\bar{K}}_\delta ^{-1}{\bar{P_r}} N_\delta (\varphi _s + \varphi _l). \end{aligned}$$

(3.52)

Given $\varphi _s$, this is a fixed point problem for $\varphi _l$. We will solve this problem in the ball $B_{l,\delta }$ defined in (3.43)). Let $\dot{H}^{0}\equiv L^2$. We begin with the following simple but key lemma

Lemma 3.4

Let Assumption [A1] hold and let $c_T:=T^{-1} e^{-\eta _0/T}\lesssim 1$ (which is weaker than Assumption [A3]). Then, for $f \in L^2(\mathbb {R}^3)$,

$$\begin{aligned}&\Vert {\bar{K}}_\delta ^{-1} f\Vert _{\dot{H}^{k-i}} \lesssim r^{-2+k} \Vert f\Vert _{\dot{H}^{-i}}, \, i\le k,\ k=0, 1 \end{aligned}$$

(3.53)

$$\begin{aligned}&\Vert {\bar{K}}_\delta ^{-1} M_\delta P_rf\Vert _{L^2} \lesssim \Vert f\Vert _{L^{2}}, \end{aligned}$$

(3.54)

$$\begin{aligned}&\Vert {\bar{K}}_\delta ^{-1} M_\delta P_rf\Vert _{\dot{H}^{1}} \lesssim \Vert f\Vert _{\delta }. \end{aligned}$$

(3.55)

Proof of Lemma 3.4

Since $-\Delta {\bar{P_r}}\ge r^2{\bar{P_r}}$, we have the inequality $r^2 \Vert f\Vert ^2$$\lesssim \langle f, {\bar{K}}_\delta f\rangle \le \Vert f\Vert \Vert {\bar{K}}_\delta f\Vert$, which gives $r^2 \Vert f\Vert \lesssim \Vert K_\delta f\Vert$, which implies (3.53) for $k=i=0$.

Since ${\bar{K}}_\delta {\bar{P_r}}\ge -\Delta {\bar{P_r}}$, we have $\Vert {\bar{P_r}} f\Vert _{\dot{H}^1}^2\lesssim \langle f, {\bar{K}}_\delta f\rangle \le \Vert {\bar{P_r}} f\Vert \Vert {\bar{K}}_\delta f\Vert$. This inequality and $\Vert {\bar{P_r}} f\Vert =\Vert \nabla ^{-1}{\bar{P_r}} \nabla f\Vert \le r^{-1} \Vert \nabla f\Vert$, where

$$\begin{aligned} \nabla ^{-1} := \nabla (-\Delta )^{-1}, \end{aligned}$$

(3.56)

give $\Vert {\bar{P_r}} f\Vert _{\dot{H}^1}\lesssim r^{-1} \Vert {\bar{K}}_\delta f\Vert$, which implies (3.53) for $k=1$.

Inequality (3.53), with $i=0$, and the bound $\Vert M_\delta \Vert \lesssim \delta ^{-2}$, proven in Proposition 3.2, yield

$$\begin{aligned}&\Vert {\bar{K}}_\delta ^{-1} M_\delta P_rf\Vert _{\dot{H}^k} \lesssim r^{k} \Vert f\Vert _{L^{2}}, \end{aligned}$$

(3.57)

for $k= 0, 1$, which for $k=0$ implies (3.54).

Finally, we prove more subtle (3.55). Using $\nabla ^{-1}$ from (3.56), we write $\nabla {\bar{K}}_\delta ^{-1} M_\delta f=\nabla {\bar{K}}_\delta ^{-1}\nabla \cdot (\nabla ^{-1} M_\delta ) f$. Proposition 3.2 shows that $\nabla {\bar{K}}_\delta ^{-1} \nabla \le 1$. It follows

$$\begin{aligned} \Vert \nabla {\bar{K}}_\delta ^{-1} M_\delta f\Vert _{L^2} \lesssim \Vert {\bar{P_r}} \nabla ^{-1} M_\delta f\Vert _{L^2}. \end{aligned}$$

(3.58)

This bound and Proposition C.4 of Appendix C imply (3.55). $\square$

Definition (3.51) and Eqs. (3.55) and (3.53), with $k=1, i=0$, show that

$$\begin{aligned} \Vert \Phi '_l\Vert _{\dot{H}^1}&\lesssim \Vert \varphi _s\Vert _{\delta }+r^{-1} \Vert \kappa '\Vert _{L^2}. \end{aligned}$$

(3.59)

For the nonlinear term, ${\bar{\Phi }}''_l(\varphi _l):= K_\delta ^{-1}{\bar{P_r}} N_\delta (\varphi _s + \varphi _l)$ (see (3.52)), Eqs. (3.53), with $k=1, i=0$, (5.1) and the inequality $\Vert \varphi _l\Vert _{s,\delta } \lesssim \Vert \varphi _l\Vert _{\dot{H}^1}$ (see (3.48)) give

$$\begin{aligned}&\Vert \Phi ''_l(\varphi _l) \Vert _{\dot{H}^1} \lesssim r^{-1} m^{-\frac{1}{3}} \delta ^{- 1/2}(\Vert \varphi _s\Vert _{\delta } + \Vert \varphi _l\Vert _{\dot{H}^1})^2. \end{aligned}$$

(3.60)

Since $\Vert \varphi _s\Vert _{ \delta } \le c_s$ and $\Vert \varphi _l\Vert _{ \delta } \le c_l$ for $\varphi _s \in B_{l,\delta }$ and $\varphi _s \in B_{s,\delta }$ (see (3.44) and (3.43)) and, due to our assumption (3.46), we have

$$\begin{aligned} \delta \Vert {\kappa '}\Vert _{L^2}&+ c_s +\delta ^{1/2} m^{-1/3} (c_s + c_l)^2\ll c_l, \end{aligned}$$

(3.61)

(3.59)–(3.61) show that $\Phi _l$ maps $B_{l,\delta }$ into itself.

Once more, by Eqs. (3.48), (3.53), with $k=1, i=0$, (3.49) and (3.55), we see that $\Phi _l$ satisfies

$$\begin{aligned}&\Vert \Phi _l(\varphi _1) - \Phi _l(\varphi _2)\Vert _{\dot{H}^{1}} \nonumber \\&\quad \lesssim r^{-1} m^{-\frac{1}{3}} \delta ^{-1/2} (\Vert \varphi _1\Vert _{{ \delta }} + \Vert \varphi _2\Vert _{{ \delta }}) \Vert \varphi _1 - \varphi _2\Vert _{{ \delta }}\, \end{aligned}$$

(3.62)

and therefore, since $r=a/\delta$, is a contraction on $B_{l,\delta }$ for $m^{-\frac{1}{3}} \delta ^{1/2}c_l\ll 1$, which follows from (3.46). Proposition 3.3 now follows by applying the fixed point theorem on $B_{l,\delta }$. $\square$

Let $\varphi _l = \varphi _l(\varphi _s)$ be the solution to Eq. (3.42) given in Proposition 3.3 with $\varphi _s \in B_{s,\delta }$. Later on we will need a Lipschitz estimate on the solution, $\varphi _l(\varphi _s) \in B_{l,\delta }$.

Lemma 3.5

If $\varphi ,\psi \in B_{s,\delta }$, then the solution, $\varphi _l(\varphi _s) \in B_{l,\delta }$, to (3.42) given in Proposition 3.3 satisfies the estimate

$$\begin{aligned} \Vert \varphi _l(\varphi )-\varphi _l(\psi )\Vert _{\dot{H}^1} \lesssim \Vert \varphi - \psi \Vert _{ \delta }. \end{aligned}$$

(3.63)

Proof

Since $\varphi _l(\varphi ), \varphi _l(\psi )$ satisfy (3.42) (and therefore (3.50)), we see that

$$\begin{aligned}&\varphi _l(\varphi ) - \varphi _l(\psi ) = -{\bar{K}}_\delta ^{-1}M_\delta (\varphi - \psi ) \nonumber \\&\quad + {\bar{K}}_\delta ^{-1} {\bar{P_r}} (N_\delta (\varphi + \varphi _l(\varphi ) ) - N_\delta (\psi + \varphi _l(\psi ))). \end{aligned}$$

(3.64)

Using Eqs. (3.64), (3.53), with $k=1, i=0$, and (3.55) and nonlinear estimate (3.49) and going through the same arguments as in the proof of Proposition 3.3, we show (3.63). $\square$

We substitute $\varphi _l = \Phi _l(\varphi _l)$ (see (3.50)), with $\Phi _l(\varphi _l)$ given by (3.50)–(3.52) into Eq. (3.41) and note that $P_r K_\delta \bar{P}_r = P_r M_\delta \bar{P}_r$ to arrive at the following equation

$$\begin{aligned} \ell \varphi _s = Q {\kappa '}+ QN(\varphi (\varphi _s)), \end{aligned}$$

(3.65)

where $\varphi (\varphi _s) = \varphi _s + \varphi _l(\varphi _s)$ with $\varphi _l(\varphi _s)$ being the solution of (3.42), and

$$\begin{aligned}&\ell := P_rK_\delta P_r- P_rM_\delta {\bar{K}}_\delta ^{-1} M_\delta P_r, \end{aligned}$$

(3.66)

$$\begin{aligned}&Q := P_r-P_rM_\delta {\bar{K}}_\delta ^{-1} . \end{aligned}$$

(3.67)

Note that $\ell$ is the Feshbach–Schur map of $K_\delta :=-\Delta + M_\delta$ with projection $P_r$.

In Sect. 3.4 below, we prove the following

Proposition 3.6

Under Assumption [A1], Eq. (3.65) has a unique solution $\varphi _s \in B_{s,\delta }$.

As a consequence of Propositions 3.3 and 3.6 and Eqs. (3.40) and (3.48), Eq. (3.31) has the unique solution $\varphi =\varphi _s + \varphi _l \in H^1 (\mathbb {R}^3)$, with the estimate

$$\begin{aligned} \Vert \varphi \Vert _{\delta } \le c_s+c_l. \end{aligned}$$

This proves the existence and uniqueness of the solution $\phi _\delta \in (H_{\mathrm{per}}^2 + H^1)(\mathbb {R}^3)$ of (3.28) (and therefore of (1.5)) with $\kappa$ given in (1.23). This completes the proof of Theorem 1.2(1). $\square$

Now, we address Theorem 1.2(2). Below, we let $\beta = T^{-1}$, so that

$$\begin{aligned} c_T=\beta e^{-\beta \eta _0}=:s_\beta . \end{aligned}$$

We begin with a result, proven in Sect. 4, which gives a detailed description of the operator $\ell$.

Proposition 3.7

On $\text {ran } P_r$, the operator $\ell$ in (3.66) is a smooth, real, even function of $-i\nabla$ and it has the expansion

$$\begin{aligned} \ell =&\nu -\nabla \epsilon \nabla + O(\delta ^2 (-i\nabla )^4) \end{aligned}$$

(3.68)

where $\nu = \delta ^{-2}|\Omega |^{-1} (m + O(s_\beta ^{2}))$, with m given in (1.33), and $\epsilon$ is a matrix given explicitly in (1.35)–(1.37) and satisfies the estimate

$$\begin{aligned} \epsilon \ge \mathbf {1}- O(s_\beta ^2) . \end{aligned}$$

(3.69)

By Proposition 3.7 the leading order term in $\ell$ is given by

$$\begin{aligned}&\ell _0 := \nu - \nabla \epsilon \nabla , \end{aligned}$$

(3.70)

where $\nu = \delta ^{-2}|\Omega | (m + O(s_\beta ^2))$, with m given in (1.33), $\epsilon \ge \mathbf {1}- O(s_\beta ^2)$.

To construct an expansion of $\varphi _s$, we let $\psi$ be the solution to the equation

$$\begin{aligned} \ell _0 \psi = \kappa ' \, \end{aligned}$$

(3.71)

(since $\nu >0$ and $\epsilon >0$, this solution exists) and write

$$\begin{aligned} \varphi _s = P_r\psi + \psi _1 \end{aligned}$$

(3.72)

where $\psi _1$ is defined by this expression. In Sect. 3.5 below to prove the following

Proposition 3.8

Under Assumption [A1], $\psi _1 \in B_{s,\delta }$ obeys the estimate

$$\begin{aligned} \Vert \psi _1\Vert _{\delta } \lesssim ( m^{1/2}+ \theta ^{1/2})\zeta . \end{aligned}$$

(3.73)

Due to (3.40) and (3.72), the solution $\varphi$ of Eq. (3.31) can be written as

$$\begin{aligned} \varphi = P_r\psi + \psi _1 + \varphi _l \, \end{aligned}$$

(3.74)

with $\psi _1 \in B_{s,\delta }$, satisfying estimate (3.73), and $\varphi _l \in B_{l, \delta }$.

To complete the proof of item (2) of Theorem 1.2, we notice that (3.30), (3.74) and the relation $P_r\psi = \psi - {\bar{P_r}} \psi$ imply (1.29) with

$$\begin{aligned} \varphi _{\mathrm{rem}} =&\psi _1- {\bar{P_r}} \psi + \varphi _l \,. \end{aligned}$$

(3.75)

Thus it remains to estimate the remainders above (see (1.30)).

Equation (3.73) controls $\psi _1$. To control the term $-{\bar{P_r}} \psi$, we use (3.56) and $\ell _0^{-1} {\bar{P_r}} \le r^{-2}$ to obtain, for $i=0, 1$,

$$\begin{aligned} \Vert {\bar{P_r}} \psi \Vert _{\dot{H}^i}&= \Vert \nabla ^i {\bar{P_r}} \ell _0^{-1} \kappa '\Vert _{L^2} = \Vert \ell _0^{-1} {\bar{P_r}} \nabla ^i \kappa '\Vert _{L^2}\nonumber \\&\lesssim r^{-2} \Vert \kappa '\Vert _{\dot{H}^i}. \, \end{aligned}$$

(3.76)

Since, by condition (1.28), $( m^{1/2}+ \theta ^{1/2})\zeta ^{2-i}\gg \delta ^{2}$, Eqs. (3.73) and (3.76), together with (3.75), show that

$$\begin{aligned} \Vert \psi _1- {\bar{P_r}} \psi \Vert _{\delta } \lesssim ( m^{1/2}+ \theta ^{1/2})\zeta . \end{aligned}$$

(3.77)

By Proposition 3.3, $\varphi _l$ is in the range of ${\bar{P_r}}$ and bounded as $\Vert \varphi _{l}\Vert _{\dot{H}^1} \lesssim c_l$. Hence, using (3.48) and taking $c_l =\omega ^{-1/4}\zeta ,\ \omega :=\max (\theta ^{2}, m)\ll 1$ (satisfying (3.46)) and using that $r^{-1}\omega \zeta = \omega m^{1/2} \zeta ^{2}\ll m^{1/4} \zeta ^{2}$, gives

$$\begin{aligned}&\Vert \varphi _{l}\Vert _{L^2} \ll m^{1/4} \zeta ^{2}, \quad \Vert \varphi _{l}\Vert _{\dot{H}^1} \lesssim \omega ^{-1/4}\zeta . \end{aligned}$$

(3.78)

By (1.34), Eqs. (3.77) and (3.78) imply part (2) of Theorem 1.2.

Finally, part (3) of Theorem 1.2 follows from Proposition 3.7 and Eqs. (3.70) and (3.71). $\square$

3.4 Small Quasi-momenta: Proof of Proposition 3.6

Our starting point is Eq. (3.65). By Proposition 3.7, the operator $\ell$ given in (3.66) is invertible. Hence we can rewrite (3.65) as the fixed point problem:

$$\begin{aligned} \varphi _s = \Phi _s(\varphi _s),\ \Phi _s(\varphi _s):= -\ell ^{-1}Q (\kappa - N(\varphi (\varphi _s))), \end{aligned}$$

(3.79)

where $\varphi (\varphi _s) = \varphi _s + \varphi _l(\varphi _s)$ with $\varphi _l(\varphi _s)$ being the solution of (3.42), and Q is given in (3.67).

First, we estimate the operator $\ell ^{-1}Q$ in $\Phi _s$. Recall, m is given in (1.33).

Lemma 3.9

Assume (3.38) and, recall, $\zeta := \delta m^{- 1/2}$. Then

$$\begin{aligned} \Vert \ell ^{-1}Qf\Vert _{ \delta } \lesssim \zeta \Vert f\Vert _{L^2}. \end{aligned}$$

(3.80)

Proof

By the choice $a := \delta r = O(1)$ (see (3.38)), we have that that

$$\begin{aligned} O(\delta ^2 (-i\nabla )^4)=O(a^2 (-i\nabla )^2)\ \text { on }\ {\text {Ran}}P_r. \end{aligned}$$

By Proposition 3.7, we have that $\nu = \delta ^{-2}|\Omega |^{-1} (m + O(s_\beta ^{2}))$, which, together with the lower bound in (1.34), implies

$$\begin{aligned} \nu > rsim \delta ^{-2}|\Omega |^{-1} m=|\Omega |^{-1} \zeta ^{-2}. \end{aligned}$$

(3.81)

These two facts and Eq. (3.68) imply $\nabla ^k\ell ^{-1} \lesssim \zeta ^{2-k},\ k=0, 1, 2$, which gives

$$\begin{aligned} \Vert \nabla ^k\ell ^{-1} \Vert \lesssim \zeta ^{2-k},\ k=0, 1, 2, \end{aligned}$$

(3.82)

for the $L^2$-operator norm. Furthermore, we claim the bound

$$\begin{aligned} \Vert \nabla ^k\ell ^{-1} P_rM_\delta {\bar{K}}_\delta ^{-1} \Vert \lesssim \zeta ^{2-k} m^{1/2} . \end{aligned}$$

(3.83)

Indeed, decomposing $M_\delta$ according to (C. 2)–(C. 4) of Proposition C.1 and using bound Eqs. (3.82), we find

$$\begin{aligned}&\Vert \nabla ^k \ell ^{-1} P_rM_\delta \nabla ^{-1}{\bar{P_r}}\Vert \nonumber \\&\quad \le \Vert \nabla ^k\ell ^{-1} P_rM_\delta ' P_r\Vert + \Vert \nabla ^k\ell ^{-1} \nabla P_r\nabla ^{-1} M_\delta '' \nabla ^{-1}{\bar{P_r}} \Vert \nonumber \\&\quad \lesssim \zeta ^{2-k}\delta ^{-1} m^{1/2} + \zeta ^{1-k} , \end{aligned}$$

(3.84)

where $\Vert \cdot \Vert$ is the operator norm in $L^2$. Since $\zeta := \delta m^{- 1/2}$, this implies

$$\begin{aligned} \Vert \nabla ^k\ell ^{-1} P_rM_\delta \nabla ^{-1}{\bar{P_r}} f\Vert _{L^2} \lesssim a^{-1} \zeta ^{1-k} \Vert f\Vert _{L^2}. \end{aligned}$$

(3.85)

Equation (3.53), with $k=1, i=0$, and (3.85), together with the insertion of $\mathbf {1}=\nabla ^{-1} \nabla =\Delta ^{-1}\nabla \nabla$ between $M_\delta$ and ${\bar{K}}_\delta ^{-1}$, imply (3.83).

Using (3.82) and (3.83) and recalling the definition $Q := P_r-P_rM_\delta {\bar{K}}_\delta ^{-1}$ (see (3.67)), we find that

$$\begin{aligned}&\Vert \nabla ^k\ell ^{-1}Q\Vert \lesssim \zeta ^{2-k}, \ k=0, 1, 2, \end{aligned}$$

(3.86)

which, due to the definition of the norm $\Vert f\Vert _{ \delta }$$\simeq \sum _0^1\zeta ^{k-1} \Vert \nabla ^k\varphi \Vert _{L^2}$ in (3.45), implies Lemma 3.9. $\square$

Lemma 3.9 and nonlinear estimate (5.1), together with $\zeta \delta ^{- 1/2}=m^{- 1/2} \delta ^{1/2}$, imply that, under Assumption [A1],

$$\begin{aligned}&\Vert \ell ^{-1}Q[N_\delta (\varphi ) - N_\delta (\psi )]\Vert _{ \delta }\nonumber \\&\qquad \qquad \lesssim m^{- 5/6}\delta ^{1/2}(\Vert \varphi \Vert _{ \delta } + \Vert \psi \Vert _{ \delta })\Vert \varphi - \psi \Vert _{ \delta }. \end{aligned}$$

(3.87)

Equation (3.79), Lemma 3.5 and estimate (3.87) imply, for $\varphi , \psi \in B_{s,\delta }$,

$$\begin{aligned}&\Vert \Phi _s(\varphi _s) \Vert _{\delta } \lesssim \zeta \Vert {\kappa '}\Vert _{L^{2}} + m^{- 5/6} \delta ^{1/2} c_s^2, \end{aligned}$$

(3.88)

$$\begin{aligned}&\Vert \Phi _s(\varphi _s) - \Phi _s(\varphi _s')\Vert _{\delta } \lesssim m^{- 5/6}\delta ^{1/2} c_s\Vert \varphi _s - \varphi _s'\Vert _{\delta }. \end{aligned}$$

(3.89)

These inequalities, together with the inequality $m^{5/6} \delta ^{- 1/2}$$=\theta ^{-3/2}\zeta \gg c_s\gg \zeta$, which follows from assumption (3.46), yield that $\Phi _s(\varphi _s)$ is a contraction on $B_{s,\delta }$ and therefore has a unique fixed point. This proves Proposition 3.6. $\square$

3.5 Proof of Proposition 3.8

In view of Proposition 3.7, we write

$$\begin{aligned}&\ell = \ell _0 + \ell ', \end{aligned}$$

(3.90)

where $\ell _0$ is defined (3.70), and $\ell '$ is defined by this expression. By Proposition 3.7, $\ell '=O(\delta ^2 (-i\nabla )^4)$ on the range of $P_r$.

Inserting (3.72) into Eq. (3.65) and using (3.90) and the relations $\ell P_r\psi = P_r{\kappa '}+ \ell ' P_r\psi$ and $- P_r{\kappa '}+ Q {\kappa '}= - P_rM_\delta {\bar{K}}_\delta ^{-1} {\kappa '}$, we obtain the equivalent equation for $\psi _1$:

$$\begin{aligned}&\ell \psi _1 =-\ell 'P_r\psi - P_rM_\delta {\bar{K}}_\delta ^{-1} {\kappa '}+ QN_\delta ({\tilde{\varphi }}) \, ,\end{aligned}$$

(3.91)

$$\begin{aligned}&{\tilde{\varphi }} = {\tilde{\varphi }}(\psi _1) := P_r\psi + \psi _1 + \varphi _l(P_r\psi +\psi _1), \end{aligned}$$

(3.92)

with $\varphi _l = \varphi _l(f)$ the solution to Eq. (3.42) given by Proposition 3.3 with $\varphi _s$ replaced by $f \in B_{s,\delta }$.

By Proposition 3.7, the operators $\ell$ and $\ell _0$ are invertible. We invert $\ell _0$ (see (3.70)) in (3.71) to obtain

$$\begin{aligned} \psi := \ell _0^{-1}{\kappa '} \, . \end{aligned}$$

(3.93)

Furthermore, we invert $\ell$ (see (3.66)) in Eq. (3.91) and use (3.95) to find

$$\begin{aligned} \psi _1 =&\Phi _1(\psi _1) = \Phi _{1}' + \Phi _1''({\tilde{\varphi }}) \, , \end{aligned}$$

(3.94)

where ${\tilde{\varphi }}$ is given in (3.92), and, with Q given in (3.67),

$$\begin{aligned}&\Phi _{1}' := -\ell ^{-1} [\ell ' \ell _0^{-1}{\kappa '}+P_rM_\delta {\bar{K}}_\delta ^{-1} {\kappa '}], \end{aligned}$$

(3.95)

$$\begin{aligned}&\Phi _1''({\tilde{\varphi }}):= \ell ^{-1}QN({\tilde{\varphi }}). \end{aligned}$$

(3.96)

(3.94) is a fixed point equation for $\psi _1$. However, we do not have to solve it since we have already proved the existence of $\psi _1$. We use (3.94) to estimate $\psi _1$.

Next, by Proposition 3.7, Eq. (3.81) and the relation $O(\delta ^2 (-i\nabla )^4)=O(a^2 (-i\nabla )^2)$ on ${\text {Ran}}P_r$ (see the definition of $P_r$ in (3.37)), valid due to the choice $a := \delta r = O(1)$ (see (3.38)), we have that the operators $\ell , \ell _0, \ell '$ given in (3.66), (3.70), and (3.90), respectively, satisfy

$$\begin{aligned}&|\ell '| \lesssim \delta ^2 (-i\nabla )^4 , \end{aligned}$$

(3.97)

$$\begin{aligned}&\ell _0 > rsim -\Delta + \zeta ^{-2}, \end{aligned}$$

(3.98)

$$\begin{aligned}&\ell > rsim -\Delta + \zeta ^{-2}, \end{aligned}$$

(3.99)

where, recall, $\zeta := m^{- 1/2} \delta$, with m given in (1.33), (cf. (3.82)).

Using (3.97)–(3.99) and the fact $\ell , \ell _0, \ell '$ are self-adjoint and are functions of $-i\nabla$ and therefore mutually commute, and using (3.83), we find that

$$\begin{aligned}&\Vert \Phi _{1}' \Vert _{\dot{H}^i} \lesssim \delta ^{ 2}\Vert {\kappa '}\Vert _{H^i}+ m^{1/2}\zeta ^{2-i}\Vert {\kappa '}\Vert _{L^2},\ i=0,1. \end{aligned}$$

(3.100)

By the choice of the $B_{s,\delta }$ norm (see (3.45)) and since $\delta ^{ 2}\ll m^{1/2}\zeta ^{2-i}$ (by (3.46), or (1.28)), we see that

$$\begin{aligned} \Vert \Phi _{1}'\Vert _{\delta } \lesssim m^{1/2} \zeta \Vert {\kappa '}\Vert _{L^2}. \end{aligned}$$

(3.101)

Now, we turn our attention to the map $\Phi ({\tilde{\varphi }}):=\ell ^{-1}QN({\tilde{\varphi }})$ (see (3.96)). The definition of $\Phi ({\tilde{\varphi }})$ and Eq. (3.87) give

$$\begin{aligned} \Vert \Phi _{1}''({\tilde{\varphi }})\Vert _{\delta } \lesssim m^{- 5/6}\delta ^{1/2}\Vert {\tilde{\varphi }}\Vert _{\delta }^2. \end{aligned}$$

(3.102)

Next, we estimate ${\tilde{\varphi }} = {\tilde{\varphi }}(\psi _1) := P_r\psi + \psi _1 + \varphi _l(P_r\psi +\psi _1)=\varphi _s+\varphi _l$ (see (3.92)). By Propositions 3.3 and 3.6, $\Vert \varphi _l\Vert _{ \delta } \le c_l$ and $\Vert P_r\psi + \psi _1\Vert _{ \delta } \lesssim c_s$ and therefore $\Vert {\tilde{\varphi }}\Vert _{ \delta } \lesssim c_s+c_l$. This, together with (3.102) and condition (3.46) and inequality (3.48), yields

$$\begin{aligned} \Vert \Phi _{1}''({\tilde{\varphi }})\Vert _{ \delta }&\lesssim m^{- 5/6}\delta ^{ 1/2}c_l^2. \end{aligned}$$

(3.103)

Equations (3.101) and (3.103) and the relation $m^{- 5/6}\delta ^{ 1/2}=\theta ^{3/2}\zeta ^{- 1}$ imply

$$\begin{aligned} \Vert \Phi _1(\psi _1)\Vert _{ \delta }&\lesssim m^{1/2} \zeta \Vert \kappa '\Vert _{L^{2}}+ \theta ^{3/2}\zeta ^{- 1} c_l^2. \end{aligned}$$

(3.104)

By condition (3.46) and our choice $c_l =\omega \zeta ,\ \omega :=\min (\theta ^{- 1/2}, m^{-1/4})$, we see that (3.104) implies

$$\begin{aligned} \Vert \Phi _1(\psi _1)\Vert _{ \delta }&\lesssim (a^{-2} m^{1/2}+ \theta ^{1/2})\zeta . \end{aligned}$$

Since $\psi _1=\Phi _1(\psi _1)$ and, the above estimate gives (3.73), proving Proposition 3.8. $\square$

4 Analysis of the Operator $\ell$. Proof of Proposition 3.7

The goal of this section is to prove Proposition 3.7. The proof follows readily from Lemmas 4.1, 4.3 and 4.4 below. Throughout this section, we suppose Assumption [A1] holds, without mentioning this explicitly.

Let $M_{ k}$ and ${\bar{K}}_{ k}$ be the k-th Bloch–Floquet fibers of $M\equiv M_{\delta =1}$ and ${\bar{K}} \equiv {\bar{K}}_{\delta =1}$ (see (2.34), not to be confused with $M_{\delta }$ and ${\bar{K}}_{\delta }$). Since, by Proposition 3.2, ${\bar{K}}$ is invertible, then so is ${\bar{K}}_{ k}$ and ${\bar{K}}_{ k}^{-1}=({\bar{K}}^{-1})_k$ (see (2.37)). We have

Lemma 4.1

The operator $\ell$, defined in (3.66), is of the form

$$\begin{aligned} \ell =&\delta ^{-2} b (-i\delta \nabla ) P_r, \end{aligned}$$

(4.1)

where b(k) is a smooth, even function of $-i\delta \nabla$ given explicitly as:

$$\begin{aligned} b (k) =&|\Omega |^{-1} \big \langle 1, (|k|^2 +M_{ k}-M_{ k} {\bar{K}}_{ k}^{-1} M_{ k}) 1 \big \rangle _{L^2_{\mathrm{per}}}. \end{aligned}$$

(4.2)

Proof

Since $M_\delta$ is $\mathcal {L}_\delta$-periodic by Proposition 3.2, Eq. (3.66) implies that so is $\ell$. Moreover, (3.66) and (3.34) yield

$$\begin{aligned} \ell =&P_r [\delta ^{-2}U_\delta (-\Delta + M) U_\delta ^*] P_r - P_r [\delta ^{-2}U_\delta M U_\delta ^*] \\&\times {\bar{P}}_r [\delta ^{-2}U_\delta (-\Delta + M) U_\delta ^*]^{-1} {\bar{P}}_r [\delta ^{-2}U_\delta M U_\delta ^*] P_r, \end{aligned}$$

where, recall, $M\equiv M_\delta \big |_{\delta = 1}$, which implies that

$$\begin{aligned} \ell = \delta ^{-2}U_{\delta }\ell \big |_{\delta = 1}U_{\delta }^* . \end{aligned}$$

(4.3)

The last two properties and Lemma 2.8 show that $\ell$ is a function of $-i\delta \nabla$ of the form (4.1), where $b(k) = \langle (\ell |_{\delta =1} )_k 1 \rangle _\Omega$, with $(\ell \mid _{\delta =1} )_k$ being the Bloch–Floquet fibres of $\ell \mid _{\delta =1}$ and 1 standing for the constant function, $1 \in L^2_{\mathrm{per}}(\mathbb {R}^3)$. Using Eqs. (2.37), (3.66) and $\Delta _k\mathbf {1}=0$, we find explicit form (4.2) of b(k).

The next proposition gives the Bloch–Floquet decomposition of the operator M.

Proposition 4.2

The operator M has a Bloch–Floquet decomposition (2.34) whose $k-$fiber, $M_{k}$, acting on $L_{\mathrm{per}}^2$ is given by

$$\begin{aligned} M_{k} f =&- {\text {den}}\left[ \oint r_{\mathrm{per, 0}}(z)f r_{\mathrm{per}, k}(z)\right] \end{aligned}$$

(4.4)

where $f \in L_{\mathrm{per}}^2$ and, on $L_{\mathrm{per}}^2$,

$$\begin{aligned}&r_{\mathrm{per, k}}(z) = (z- h_{\mathrm{per, k}})^{-1}, \end{aligned}$$

(4.5)

$$\begin{aligned}&h_{\mathrm{per, k}} = (-i\nabla - k)^2 - \phi _{\mathrm{per}} \, . \end{aligned}$$

(4.6)

Proof of Proposition 4.2

Let $T_s$ be given in (2.1) and $\varphi \in L^2$. To compute k-fibers of M, we note $T_{-t}{\text {den}}[A] = {\text {den}}\left[ T_t^* A T_t \right]$ and $[ T_t, r_{\mathrm{per}}(z)]=0$ for all $t \in \mathcal {L}$. Using these relations, the definition of the Bloch–Floquet decomposition (2.23) and Eq. (3.11), we obtain

$$\begin{aligned} (M \varphi )_k(x)&= - \sum _{t \in \mathcal {L}} e^{- ik(x+t)} \oint T_{-t}{\text {den}}\left[ r_{\mathrm{per}}(z) \varphi r_{\mathrm{per}}(z) \right] \nonumber \\&= - \sum _{t \in \mathcal {L}} e^{- ik(x+t)}\oint {\text {den}}\left[ T_t^* r_{\mathrm{per}}(z) \varphi r_{\mathrm{per}}(z) T_t \right] . \end{aligned}$$

(4.7)

Since $r_{\mathrm{per}}(z)$ is $\mathcal {L}$-periodic, (4.7) shows

$$\begin{aligned}&(M \varphi )_k (x) \nonumber \\&\quad = - \sum _{t \in \mathcal {L}} e^{-2\pi ik(x+t)}\oint {\text {den}}\left[ r_{\mathrm{per}}(z) (T_{-t}\varphi ) r_{\mathrm{per}}(z) \right] . \end{aligned}$$

(4.8)

Using that ${\text {den}}[A]f = {\text {den}}[Af] = {\text {den}}[fA]$ for any operator A on $L^2(\mathbb {R}^3)$ and any sufficiently regular function f on $\mathbb {R}^3$, we insert the constant factor of $e^{-ikt}$ into ${\text {den}}$ in (4.8). We obtain

$$\begin{aligned}&(M \varphi )_k(x) \nonumber \\&\quad = - e^{-ikx} \oint {\text {den}}\left[ r_{\mathrm{per}}(z) \sum _{t \in \mathcal {L}} e^{- ikt} (T_{-t}\varphi ) r_{\mathrm{per}}(z) \right] . \end{aligned}$$

(4.9)

This and the definition of the Bloch–Floquet decomposition of $\varphi$, (2.23), imply

$$\begin{aligned} (M \varphi )_k(x) =&- \oint {\text {den}}\left[ r_{\mathrm{per}}(z) \varphi _k e^{ ikx} r_{\mathrm{per}}(z) e^{-ikx} \right] . \end{aligned}$$

(4.10)

Since $e^{ ikx} (-i\nabla ) e^{- ikx} = -i\nabla - k$, and therefore $e^{ i kx}r_{\mathrm{per}}(z)e^{- ikx} = r_{\mathrm{per}, k}(z)$, this gives (4.4). $\square$

Since the resolvents $r_{\mathrm{per, k}}(z)$ are smooth in k (see (4.5)–(4.6)), then, by (4.4), $M_{ k}$ is also smooth in k. Hence, by (4.2), b(k) is smooth in k.

Since the operator $M_{ k}-M_{ k} {\bar{K}}_{ k}^{-1} M_{ k}$ in (4.2) is self-adjoint, the function b(k) is real. By Lemma 4.2 and the properties ${\bar{r}}_{\mathrm{per, k}}(z) := \mathcal {C}r_{\mathrm{per, k}}(z)\mathcal {C}$$= r_{\mathrm{per, - k}}({\bar{z}})$, where $\mathcal {C}$ is the complex conjugation, and the contour of integration in (4.4) is symmetric w.r.to the reflection $z\rightarrow {\bar{z}}$, we have $b(k)={\bar{b}}(k)=b(- k)$, i.e. b(k) is even. $\square$

Let $K_{ 0} = K_{ k=0}$ denote the 0-fiber of K, acting on $L^2_{\mathrm{per}}(\mathbb {R}^3)$. We also let $\Pi _0$ denote the projection onto constant functions on $L^2_{\mathrm{per}}(\mathbb {R}^3)$ and ${\bar{\Pi }}_0 := 1- \Pi _0$. Finally, we define

$$\begin{aligned} {\bar{K}}_{ 0} := {\bar{\Pi }}_0 K_{ 0} {\bar{\Pi }}_0. \end{aligned}$$

(4.11)

Recall the abbreviation $s_\beta :=\beta e^{- \eta _0\beta }$. With this notation, we have

Lemma 4.3

Let m be given in (1.33). The function b(k) given in (4.2) satisfies

$$\begin{aligned} b(k)&= |\Omega |^{-1} (m + O(s_\beta ^2)) + k \cdot \epsilon k \nonumber \\&\quad + k \cdot O(s_\beta ) k +O(|k|^4), \end{aligned}$$

(4.12)

where m is a scalar given by (1.33) and $\epsilon$ is a real matrix given by (1.35)–(1.37).

Proof of Lemma 4.3

First, we use (4.2) to write b(k) as

$$\begin{aligned}&b (k) = b_1(k) - b_2(k), \end{aligned}$$

(4.13)

$$\begin{aligned}&b_1(k) := |k|^2 + |\Omega |^{-1} \left\langle 1, M_{ k} 1 \right\rangle _{L^2_{\mathrm{per}}}, \end{aligned}$$

(4.14)

$$\begin{aligned}&b_2(k) := |\Omega |^{-1} \langle 1, M_{ k} {\bar{K}}_{ k}^{-1} M_{ k} 1\rangle _{L^2_{\mathrm{per}}}. \end{aligned}$$

(4.15)

We begin with $b_1(k)$. We claim that

$$\begin{aligned} b_1(k) =&|\Omega |^{-1} m + \epsilon '_1 |k|^2 + k \cdot (1 + \epsilon ') k + O(|k|^4), \end{aligned}$$

(4.16)

where m and $\epsilon '$ are given in (1.33) and (1.36), respectively, and $\epsilon '_1$ is a real, symmetric matrix satisfying $\epsilon '_1 = O(s_\beta ),$ contributing to the third term on the r.h.s. of (4.12).

Using definition of $b_1$ in (4.14) and Proposition 4.2, we see that

$$\begin{aligned} b_1(k)&= - |\Omega |^{-1}\langle 1, {\text {den}}\oint r_{\mathrm{per}, 0}(z) 1 r_{\mathrm{per}, k}(z) \rangle _{L^2(\Omega )} \end{aligned}$$

(4.17)

$$\begin{aligned}&= - |\Omega |^{-1} \mathrm {Tr}_{\Omega } \oint r_{\mathrm{per}, 0}(z) r_{\mathrm{per}, k}(z), \end{aligned}$$

(4.18)

where 1 is the constant function $1 \in L_{\mathrm{per}}^2$ and $\Omega$ is an arbitrary fundamental cell of $\mathcal {L}$. To begin with, using the Cauchy-formula for derivatives, we obtain

$$\begin{aligned} b_1(0) =&- |\Omega |^{-1} \mathrm {Tr}_{\Omega } \oint r_{\mathrm{per}, 0}^2(z) =|\Omega |^{-1} m. \end{aligned}$$

(4.19)

Next, recall that $h_{\mathrm{per}, k} = (-i\nabla - k)^2 - \phi _{\mathrm{per}}$ (see (4.6)). We have, by the resolvent identity, that

$$\begin{aligned} r_{\mathrm{per}, k}(z)&- r_{\mathrm{per}, 0}(z) = r_{\mathrm{per}, k}(z) [2(-i\nabla ) \cdot k - |k|^2] r_{\mathrm{per}, 0}(z). \end{aligned}$$

(4.20)

Applying this identity to (4.18) and using that $b_1(k)$ is even, we obtain (4.16), with $\epsilon '_1:= - |\Omega |^{-1} \mathrm {Tr}_{L^2_{\mathrm{per}}} \oint r_{\mathrm{per}, 0}^3(z)$ and $\epsilon '$ given by (1.36).

Using the Cauchy-integral formula, we rewrite $\epsilon '_1$ as

$$\begin{aligned} \epsilon '_1 =&-\frac{1}{2} \mathrm {Tr}_{L^2_{\mathrm{per}}} f''_{T}(h_{\mathrm{per}}- \mu ). \end{aligned}$$

(4.21)

Then following the proof of Lemma B.2 with $f'_{\mathrm{FD}}$ replaced by $f''_{\mathrm{FD}}$, we show that $\epsilon '_1 = O(s_\beta )$. This proves (4.16).

Next, we prove the expansion

$$\begin{aligned} b_2(k) = k \cdot \epsilon '' k + k \cdot \epsilon _1'' k +O(|k|^4) + O(s_\beta ^2), \end{aligned}$$

(4.22)

where $\epsilon ''$ is given in (1.37), $\epsilon _1''$ is a real matrix satisfying $\epsilon _1'' = O(s_\beta )$ (contributing to the third term on the r.h.s. of (4.12)). First, we recall from (4.15)

$$\begin{aligned} b_2(k)&= |\Omega |^{-1}\langle 1, (M {\bar{K}}^{-1} M)_k 1 \rangle _{L^2_{\mathrm{per}}} \end{aligned}$$

(4.23)

$$\begin{aligned}&= |\Omega |^{-1}\langle M_{ k} 1, {\bar{K}}_{ k}^{-1} M_{ k} 1 \rangle _{L^2_{\mathrm{per}}} , \end{aligned}$$

(4.24)

where, recall, $M_{ k}, K_{ k}$ and ${\bar{K}}_{ k}$ are the k-th Bloch–Floquet fiber of $M\equiv M_{\delta =1}, K\equiv K_{\delta =1}$ and ${\bar{K}}\equiv {\bar{K}}_{\delta =1}$. Letting

$$\begin{aligned} \rho _k = ({\bar{P}}_a)_k M_{ k} 1 \in L_{\mathrm{per}}^2\, , \end{aligned}$$

(4.25)

where $L_{\mathrm{per}}^2$ is given in (1.19), a is given in (3.38), and $P_a$ is defined in (3.37), we find

$$\begin{aligned} b_2(k) = |\Omega |^{-1} \langle \rho _k, {\bar{K}}_{ k}^{-1} \rho _k \rangle _{L^2_{\mathrm{per}}} \, . \end{aligned}$$

(4.26)

Now, we expand $\rho _k$ in k. By (4.4), we have $M_{k=0} \mathbf {1}$$= - {\text {den}}\left[ \oint r_{\mathrm{per, 0}}^2(z)\right]$. Next, recall $\oint := \frac{1}{2\pi i} \int _\Gamma dz f_{T}(z-\mu )$ (see (3.7)) to obtain

$$\begin{aligned} M_{k=0} \mathbf {1}=- {\text {den}}f_{T}'(h_{\mathrm{per}, 0}-\mu ). \end{aligned}$$

(4.27)

Since $f_{FD}'\le 0$, we have $M_{k=0} \mathbf {1}>0$. Introduce the function

$$\begin{aligned}&V(x) = - {\text {den}}\left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] (x) \ge 0. \end{aligned}$$

(4.28)

By definition (4.25) and Eqs. (4.4), (4.27) and (4.28), we have

$$\begin{aligned}&\rho _k = ({\bar{P}}_a)_k V + \rho '_k, \end{aligned}$$

(4.29)

$$\begin{aligned}&\rho '_k := ({\bar{P}}_a)_k {\text {den}}\oint r_{\mathrm{per}, 0}^2(z)(2(-i\nabla ) k + k^2) r_{\mathrm{per}, k}(z). \end{aligned}$$

(4.30)

Inserting the decomposition (4.29) into (4.26) gives

$$\begin{aligned} |\Omega |b_2(k) =&\langle V, {\bar{K}}_{ k}^{-1} V \rangle + 2\mathrm {Re}\langle V, {\bar{K}}_{ k}^{-1} \rho _{k}' \rangle + \langle \rho _k', {\bar{K}}_{ k}^{-1} \rho _{k}' \rangle . \end{aligned}$$

(4.31)

We expand the third term on the r.h.s. on (4.31). First, we give a rough bound. For $z\in \Gamma$, we claim the estimates

$$\begin{aligned}&\Vert (1-\Delta )^{\alpha } r_{\mathrm{per}, k} (z)\Vert \nonumber \\&\quad \le \ \Vert (1-\Delta )^{\alpha } r_{\mathrm{per}}(z)\Vert \lesssim \ d^{\alpha -1}\lesssim 1, \end{aligned}$$

(4.32)

for $\alpha =0, 1/2$, where $d \equiv d(z):=\mathrm {dist}(z, \sigma (h_{\mathrm{per}}))\ge \frac{1}{4} .$ The first estimate follows from (2.38). The second estimate is straightforward for $\alpha =0, 1$, which by interpolation, gives it for all $\alpha \in [0, 1]$. For $\alpha = 1/2$, it can be also proven directly as $\Vert (-\Delta )^{1/2}f\Vert ^2=\langle f, (h_{\mathrm{per}}-z+\phi _{\mathrm{per}}+z) f\rangle$. Taking $f = r_{\mathrm{per}}(z)u$, we arrive at the second estimate in (4.32) for $\alpha =1/2$.

By the second resolvent identity (4.20) and estimates (4.32), we have the expansion

$$\begin{aligned} r_{\mathrm{per}, k}(z)=r_{\mathrm{per}, 0}(z)+O(|k| d^{-3/2} +|k|^2 d^{-2}). \end{aligned}$$

Using this expansion in (4.30), we find

$$\begin{aligned} \rho _{k}' =\rho ' \cdot k +O(|k|^2), \end{aligned}$$

where $\rho '$ is given in (1.38). Using the latter relation, the relation ${\bar{K}}_{ k}^{-1}={\bar{K}}_{ 0}^{-1}+O(k)$ and the fact that, since on $L^2_{\mathrm{per}}$ the spectrum of $-i\nabla$ is discrete, $({\bar{P}}_a)_{k=0} = ({\bar{P}}_{a=0})_{k=0}$ for a is sufficiently small, we obtain

$$\begin{aligned} |\Omega |^{-1}\langle \rho _k', {\bar{K}}_{ k}^{-1} \rho _k' \rangle = - k \epsilon '' k + O(k^4), \end{aligned}$$

(4.33)

for $\epsilon ''$ is given in (1.37), where the power of the remainder comes from the fact $b_2(k)$ is even which is shown by the same argument that was used in demonstration that b(k) is even. Equations (4.31) and (4.33) show that

$$\begin{aligned}&b_2(k) = b_2(0) - k \epsilon '' k + O(k^4) + \text {Rem}, \end{aligned}$$

(4.34)

$$\begin{aligned}&\text {Rem} := \langle V, [{\bar{K}}_{ k}^{-1} - {\bar{K}}^{-1}_0]V \rangle + 2 \mathrm {Re}\langle V, {\bar{K}}_{ k}^{-1} \rho _{ k}' \rangle , \end{aligned}$$

(4.35)

with $b_2(0):=|\Omega |^{-1}\langle V, {\bar{K}}_{ k}^{-1} V \rangle$. To estimate $b_2(0)$ and the terms in (4.35), we use Eq. (3.53) and the relation $\Vert {\bar{K}}^{-1}\Vert =\sup _k \Vert {\bar{K}}_{ k}^{-1}\Vert$ (see (2.38)) to obtain

$$\begin{aligned} \Vert {\bar{K}}_{ k}^{-1}\Vert \lesssim 1. \end{aligned}$$

We use this bound, Lemma B.2, (4.30) and the fact that $b_2(k)$ is even in k, to obtain

$$\begin{aligned} |\Omega | |\text {Rem}|&\lesssim (\Vert V\Vert _{L^2_{\mathrm{per}}}^2 + \Vert V\Vert _{L^2_{\mathrm{per}}})|k|^2\nonumber \\&= O(s_\beta |k|^2), \end{aligned}$$

(4.36)

$$\begin{aligned} |\Omega | b_2(0)&= O(\Vert V\Vert _{L^2_{\mathrm{per}}}^2) = O(s_\beta ^2). \end{aligned}$$

(4.37)

We identify the first, third and fourth terms on the r.h.s. of (4.34) with the fourth, third and second terms in (4.22), respectively. Equations (4.34)–(4.36) imply (4.22).

Equations (4.13), (4.16), and (4.22) yield equation (4.12), with $\epsilon _1'+\epsilon _1''$ making up the third term on the r.h.s. of (4.12). This completes the proof of Lemma 4.3. $\square$

Lemma 4.4

The $3 \times 3$ matrix $\epsilon$ entering (4.2) is symmetric and satisfies

$$\begin{aligned} \epsilon \ge \mathbf {1}- O(s_\beta ^2) . \end{aligned}$$

(4.38)

Proof

We prove this lemma using the Feshbach–Schur map. Let $P = P_s$ (see (3.37)) for some real number $s > 0$, unrelated to r and satisfying $B(\delta s)\subset \Omega ^*$. For any projection P and operator H on $L^2(\mathbb {R}^3)$, the Feshbach–Schur map $F_P(H)$ is defined as

$$\begin{aligned} F_P(H) := P H P - PH {\bar{P}} {\bar{H}}^{-1} {\bar{P}} H P. \end{aligned}$$

(4.39)

where ${\bar{P}} = 1-P$, ${\bar{H}} = {\bar{P}} H {\bar{P}}$, and ${\bar{H}}^{-1}$ is defined on the range of ${\bar{P}}$. The Feshbach–Schur map has the property [20]

$$\begin{aligned} -\lambda \notin \sigma (H) \iff -\lambda \notin \sigma (F_P(H+\lambda ) - \lambda P). \end{aligned}$$

(4.40)

for any $\lambda \ge 0$. That is, for all $\lambda > 0$,

$$\begin{aligned} H \ge 0 \iff F_P(H+\lambda ) - \lambda P \ge 0. \end{aligned}$$

(4.41)

With the Laplacian $\Delta$, we define

$$\begin{aligned} K_{c, \delta } = K_\delta + c \Delta . \end{aligned}$$

(4.42)

Since $M_\delta > 0$ by Proposition 3.1, we have that $K_{c, \delta } > 0$ for all $c \in [0,1)$. Consequently, (4.41) shows that, for any $\lambda > 0$,

$$\begin{aligned} F_P(K_{c, \delta } + \lambda ) - \lambda P \ge 0. \end{aligned}$$

(4.43)

Using definition (4.39) and the resolvent identity, we obtain

$$\begin{aligned}&F_P (K_{c, \delta } + \lambda ) - \lambda P \end{aligned}$$

(4.44)

$$\begin{aligned}&\quad = PK_{c, \delta } P - PM_\delta ({\bar{K}}_{c, \delta } + \lambda {\bar{P}})^{-1} M_\delta P \end{aligned}$$

(4.45)

$$\begin{aligned}&\quad = F_P(K_{c, \delta }) + \lambda PM_\delta {\bar{K}}_{c, \delta }^{-1} ({\bar{K}}_{c, \delta } + \lambda {\bar{P}})^{-1} M_\delta P \end{aligned}$$

(4.46)

$$\begin{aligned}&\quad = F_P(K_{c, \delta }) + \lambda PM_\delta {\bar{K}}_{c, \delta }^{-2} M_\delta P \nonumber \\&\qquad - \lambda ^2 PM_\delta ({\bar{K}}_{c, \delta })^{-2}({\bar{K}}_{c, \delta } + \lambda {\bar{P}})^{-1} M_\delta P. \end{aligned}$$

(4.47)

By the choice of $P = P_s$ (see (3.37)), we see that ${\bar{K}}_{c, \delta } > rsim s^2$. Since $M_\delta \lesssim \delta ^{-2}$, we see that the last term in (4.47) is bounded by $O(\lambda ^2 \delta ^{-4} s^{-6})$. Thus, (4.44) - (4.47) implies

$$\begin{aligned} F_P(K_{c, \delta } + \lambda ) - \lambda P&= F_P(K_{c, \delta }) + \lambda W+ O(\lambda ^2 \delta ^{-4} s^{-6})P, \end{aligned}$$

(4.48)

where $W:=PM_\delta ({\bar{K}}_{c, \delta })^{-2} M_\delta P$. To estimate W, we proceed as in the proof of Lemma 4.1. First, since $M_\delta$ is $\mathcal {L}_\delta$-periodic by Proposition 3.2, W is $\mathcal {L}_\delta$-periodic. Moreover, the definition $W:=PM_\delta ({\bar{K}}_{c, \delta })^{-2} M_\delta P$ and (3.34) yield

$$\begin{aligned} W&= P_s (\delta ^{-2}U_\delta M_1 U_\delta ^*) {\bar{P}}_s [\delta ^{-2}U_\delta (-\Delta + M_1) U_\delta ^*]^{-2} {\bar{P}}_s (\delta ^{-2}U_\delta M_1 U_\delta ^*) P_s \ \\&= P_{\delta s} M_1 {\bar{P}}_{\delta s} {\bar{K}}_1^{-1} {\bar{P}}_{\delta s} M_1 P_{\delta s}, \end{aligned}$$

which implies that

$$\begin{aligned} W = U_{\delta }W\big |_{\delta = 1}U_{\delta }^* . \end{aligned}$$

(4.49)

($({\bar{K}}_{c, \delta })^{-1}$ entering W in the second power eats up $\delta ^{-2}$ compared to (4.3).) Since $B(\delta s)\subset \Omega ^*$, the last two properties and Lemma 2.8 show that W is a function of $-i\delta \nabla$ of the form

$$\begin{aligned} W =&w (-i\delta \nabla ) P, \end{aligned}$$

(4.50)

where $w(k) = \langle W_k 1 \rangle _\Omega$, with $W_k$ being the Bloch–Floquet fibers of W and 1 standing for the constant function, $1 \in L^2_{\mathrm{per}}(\mathbb {R}^3)$. Using Eq. (2.37), we find, as in (4.1)–(4.2), the explicit form of w(k):

$$\begin{aligned} w (k) =&|\Omega |^{-1} \big \langle 1, M_{ k} {\bar{K}}_{ k}^{-2} M_{ k} 1 \big \rangle _{L^2_{\mathrm{per}}}, \end{aligned}$$

(4.51)

where $M_{ k}$ and ${\bar{K}}_{ k}$ are the k-th Bloch–Floquet fibres of $M_{\delta =1}$ and ${\bar{K}}_{\delta =1}$.

Since the operator $M_{ k} {\bar{K}}_{ k}^{-2} M_{ k}$ in (4.2) is self-adjoint, the function w(k) is real. Arguing as with b(k) in the proof of Lemma 4.1, we conclude that w(k) is even and smooth. Furthermore, as with $b_2(k)$ in the proof of Lemma 4.3, we expand w(k) in k to the fourth order to obtain

$$\begin{aligned} W&=O(s_\beta ^2) -\delta ^2 \nabla \epsilon _3 \nabla P + O(\delta ^4 (-i\nabla )^4P), \end{aligned}$$

(4.52)

$$\begin{aligned} \epsilon _3&:= |\Omega |^{-1} \langle \rho ', {{\bar{K}}_{c, 0}}^{-2} \rho '\rangle _{L^2_{\mathrm{per}}} > 0, \end{aligned}$$

(4.53)

where $\rho '$ is given in (1.38), $K_{c, 0}\equiv K_{c, \delta =1, k= 0}$ is the 0-th fiber of $K_{c, \delta =1}$, and ${\bar{K}}_{c, 0} = {\bar{\Pi }}_0 K_{c, 0} {\bar{\Pi }}_0$. Here ${\bar{\Pi }}_0 = 1 - \Pi _0$ and $\Pi _0$ is the projection in $L^2_{\mathrm{per}}$ onto constants. The inverse ${\bar{K}}_{c, 0}^{-2}$ is taken on the range of ${\bar{\Pi }}_0$. Equations (4.48) and (4.52) imply that

$$\begin{aligned} F_P(K_{c,\delta } + \lambda ) - \lambda P&= F_P(K_{c, \delta }) +O(s_\beta ^2)- \lambda \delta ^2 \nabla \epsilon _3 \nabla P\nonumber \\&\quad + O(\delta ^4 (-i\nabla )^4 P)+O(\lambda \delta ^{-4} s^{-6} P). \end{aligned}$$

(4.54)

Now, we use definition (4.42) to expand the term $F_P(K_{c, \delta })$ in (4.54) in c. A simple computation shows that

$$\begin{aligned} F_P(K_{c, \delta })&= F_P(K_\delta ) + c \Delta P \end{aligned}$$

(4.55)

$$\begin{aligned}&\quad - \sum _{n \ge 1} c^n PM_\delta ({\bar{K}}_{\delta }^{-1} (-\Delta ))^n {\bar{K}}_{\delta }^{-1} M_\delta P. \end{aligned}$$

(4.56)

Since ${\bar{K}}_{\delta }\ge 0$, (4.56) is negative, we conclude

$$\begin{aligned} F_P(K_{c, \delta })&\le F_P(K_\delta ) + c \Delta P . \end{aligned}$$

(4.57)

Since $F_P(K_\delta ) = \ell$ for $r=s$ (see (3.66)), we see, by Lemma 4.3, that

$$\begin{aligned} F_P(K_\delta ) =&-\nabla \epsilon \nabla P + O(\delta ^2 (-i\nabla )^4 P, \end{aligned}$$

(4.58)

with $\epsilon$ defined there. We use that $O(\delta ^4 (-i\nabla )^4 P) = O({\tilde{a}}^2 (-i\nabla )^2 P)$, where ${\tilde{a}} := \delta s$ (which is unrelated to the a in (3.38)) and (4.57) and (4.58) to obtain

$$\begin{aligned} F_P(K_{c, \delta })&\le -\nabla (\epsilon - c + O({\tilde{a}}^2))\nabla P. \end{aligned}$$

(4.59)

Setting $\epsilon _4:=O({\tilde{a}}^2)+ \lambda \delta ^2\epsilon _3$, we see that Eqs. (4.54), (4.43) and (4.59) imply

$$\begin{aligned}&-\nabla (\epsilon + \epsilon _4 - c)\nabla P +O(\lambda \delta ^{-4} s^{-6}) P+O(s_\beta ^2) P \nonumber \\&\quad \ge F_P(K_{c,\delta } + \lambda ) - \lambda P \ge 0. \end{aligned}$$

(4.60)

Inequality (4.60) holds for all $s \in (0, \delta ^{-1})$. Taking $s=\delta ^{-3/4}$, we find

$$\begin{aligned} -\nabla \epsilon \nabla P \ge [(c-O(\delta ^{1/2}))\Delta -O(\lambda \delta ^{1/2}) -O(s_\beta ^2)] P_\delta , \end{aligned}$$

where $P_\delta :=P_{s=\delta ^{-3/4}}$. Since this holds for every $\delta >0$, since $P\equiv P_s$ converges strongly to $\mathbf {1}$, as $s\rightarrow \infty$, and since the expression for $\epsilon$ given in Lemma 4.3 is independent of $\delta$, we see that

$$\begin{aligned} -\nabla \epsilon \nabla \ge - c \Delta - O(s_\beta ^2), \end{aligned}$$

for every $c \in [0,1)$. Passing to the Fourier transform gives $\xi \cdot \epsilon \xi \ge c |\xi |^2 - O(s_\beta ^2), \forall \xi \in \mathbb {R}^3$. For $\xi \in \mathbb {R}^3,$ with $|\xi |\ge 1$, this implies $\xi \cdot \epsilon \xi \ge (c - O(s_\beta ^2)) |\xi |^2$, which is equivalent to (4.38). $\square$

5 Nonlinear Estimates

Let $N_\delta$ be given implicitly by (3.31) and recall the definition of the $B_{s,\delta }$ norm from (3.45). Let $\dot{H}^{0}\equiv L^2$. In this section we prove estimates on $N_\delta$.

Proposition 5.1

Let Assumption [A1] hold. If $\Vert \varphi _1\Vert _{B_{s,\delta }}, \Vert \varphi _2\Vert _{B_{s,\delta }} = o(\delta ^{-1/2})$, then we have the estimate

$$\begin{aligned}&\Vert N_\delta (\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2} \nonumber \\&\quad \lesssim m^{ -1/3} \delta ^{-1/2}(\Vert \varphi _1\Vert _{B_{s,\delta }} + \Vert \varphi _2\Vert _{B_{s, \delta }})\Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^1}. \end{aligned}$$

(5.1)

In Appendix D, we prove a more refined estimate. We derive Proposition 5.1 from its version with $\delta = 1$ by rescaling. For $\delta = 1$, we have the following result.

Proposition 5.2

Let Assumption [A1] hold and either $\Vert \psi \Vert _{L^2} = o(1)$ or $\Vert \nabla \psi \Vert _{L^2} = o(1)$. Then $N:=N_{\delta =1}$ satisfies the estimate

$$\begin{aligned}&\Vert N(\psi _1) - N(\psi _2)\Vert _{L^2}\lesssim \sum _{j=1}^2\bigg [ (\Vert \psi _j\Vert _{\dot{H}^1})\Vert \psi _1 - \psi _2\Vert _{\dot{H}^1}\nonumber \\&\quad + \big (\Vert \psi _j\Vert _{\dot{H}^1}^{1/3} \Vert \psi _j\Vert _{L^2}^{2/3} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}}\nonumber \\&\quad + \Vert \psi _j\Vert _{\dot{H}^{1}} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}}^{1/3}\Vert \psi _1 - \psi _2\Vert _{L^2}^{2/3}\big )\bigg ]. \end{aligned}$$

(5.2)

We first derive Proposition 5.1 from Proposition 5.2 and then prove the latter statement.

Proof of Proposition 5.1

By (3.33), $N_\delta$ and the unscaled nonlinearity $N = N_{\delta = 1}$ are related via

$$\begin{aligned} N_\delta (\varphi ) = \delta ^{-3/2} U_{\delta }N( \psi ),\ \qquad \psi = \delta ^{-1/2} U_{\delta }^* \varphi , \end{aligned}$$

(5.3)

where $U_{\delta }$ is given in (2.45). Equations (5.2) and (5.3) the relation $\Vert U_{\delta }^* \varphi \Vert _{L^2}= \Vert \varphi \Vert _{L^2}$ and the notation $\psi _j = \delta ^{-1/2} U_{\delta }^* \varphi _j$ imply

$$\begin{aligned} \Vert N_\delta (\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2}&\lesssim \delta ^{-3/2} \sum _{j=1}^2\bigg [\Vert \psi _j\Vert _{\dot{H}^1}\Vert \psi _1 - \psi _2\Vert _{\dot{H}^1} \nonumber \\&\quad + \Vert \psi _j\Vert _{\dot{H}^1}^{1/3} \Vert \psi _j\Vert _{L^2}^{2/3} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}} \nonumber \\ {}&\quad + \Vert \psi _j\Vert _{\dot{H}^{1}} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}}^{1/3}\Vert \psi _1 - \psi _2\Vert _{L^2}^{2/3} \bigg ].\end{aligned}$$

(5.4)

Furthermore, using the relation $\Vert \psi _j\Vert _{\dot{H}^{k}}= \delta ^{-1/2}\Vert U_{\delta }^* \varphi _j\Vert _{\dot{H}^{k}}= \delta ^{k-1/2}\Vert \varphi \Vert _{\dot{H}^{k}}$, we find

$$\begin{aligned} \Vert N_\delta (\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2}&\lesssim \delta ^{- 3/2} \sum _{j=1}^2\bigg [\delta \Vert \varphi _j\Vert _{\dot{H}^1} \Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^1} \nonumber \\&\quad + \delta ^{\frac{1}{3}} \big (\Vert \varphi _j\Vert _{\dot{H}^1}^{1/3} \Vert \varphi _j\Vert _{L^2}^{2/3} \Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^{1}} \nonumber \\&\quad + \Vert \varphi _j\Vert _{\dot{H}^{1}} \Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^{1}}^{1/3}\Vert \varphi _1 - \varphi _2\Vert _{L^2}^{2/3}\big )\bigg ] . \end{aligned}$$

(5.5)

To estimate the terms on the r.h.s. of (5.5) we use the inequality $a^{1/3} b^{2/3}\le \frac{2}{3} (a+b)$, with $a:=\Vert \varphi \Vert _{\dot{H}^1}$ and $b:= m^{ 1/2}\delta ^{-1}\Vert \psi \Vert _{L^2}$, to obtain

$$\begin{aligned} \Vert \varphi \Vert _{\dot{H}^1}^{1/3} \Vert \psi \Vert _{L^2}^{2/3}\le \frac{2}{3} (m^{ -1/2}\delta )^{2/3}(\Vert \varphi \Vert _{\dot{H}^1}+m^{ 1/2}\delta ^{-1}\Vert \psi \Vert _{L^2}). \end{aligned}$$

With the definition of the norm $\Vert \cdot \Vert _{\delta }$ in (3.45), this yields $\delta ^{\frac{1}{3}} \Vert \varphi \Vert _{\dot{H}^1}^{1/3} \Vert \varphi \Vert _{L^2}^{2/3} \Vert \chi \Vert _{\dot{H}^{1}}$

$\le \frac{2}{3} m^{ -1/3}\delta \Vert \varphi \Vert _{\delta } \Vert \chi \Vert _{\dot{H}^{1}}.$ Since $\Vert \chi \Vert _{\dot{H}^{1}}\le \Vert \chi \Vert _{\delta }$, this in turn implies

$$\begin{aligned} \delta ^{\frac{1}{3}} \Vert \varphi \Vert _{\dot{H}^1}^{1/3} \Vert \varphi \Vert _{L^2}^{2/3} \Vert \chi \Vert _{\dot{H}^{1}}\le \frac{2}{3} m^{ -1/3}\delta \Vert \varphi \Vert _{\delta }\Vert \chi \Vert _{\delta }. \end{aligned}$$

Applying this inequality to (5.5), we arrive at (5.1). $\square$

Proof of Proposition 5.2

Let $h_{\mathrm{per}}$ and $r_{\mathrm{per}}(z)$ be given in (3.8). First we observe that Eqs. (3.28)–(3.32), with $\delta =1$, read

$$\begin{aligned} N (\psi )&= F(\phi )- F(\phi _{\mathrm{per}})-d_\varphi F (\phi _{\mathrm{per}})\psi , \end{aligned}$$

(5.6)

$$\begin{aligned} F (\phi )&= {\text {den}}[ f_T(h^\phi - \mu )], \end{aligned}$$

(5.7)

where $\psi := \phi - \phi _{\mathrm{per}}$ and, recall, $h^\phi :=- \Delta - \phi =h_{\mathrm{per}}- \psi$. Next, using Eqs. (3.4) and (3.7) and expanding $(z-h^{\phi })^{-1}=(z-h_{\mathrm{per}}+ \psi )^{-1}$ to the second order, we find

$$\begin{aligned} N(\psi ) := {\text {den}}[\tilde{N}_2(\psi )], \end{aligned}$$

(5.8)

where

$$\begin{aligned} \tilde{N}_k(\psi ) := \oint (z-h_{\mathrm{per}}+ \psi )^{-1} [(-\psi ) r_{\mathrm{per}}(z) ]^k , \end{aligned}$$

(5.9)

with $\oint$ given by $\oint := \frac{1}{2\pi i} \int _\Gamma dz f_{T}(z-\mu )$, where $\Gamma$ is the contour given in Fig. 1 (see (3.7)), equipped with the positive orientation.

We deform the contour $\Gamma$ given in Fig. 1 into the contour indicated in Fig. 2 by the blue dashed line and consisting of two separate contours traversed counter-clockwise.

By the formal resolvent expansion (without justifying the convergence)

$$\begin{aligned} (z-h_{\mathrm{per}}+\psi )^{-1} =\sum _{k=2}^{\infty } r_{\mathrm{per}}(z) [(-\psi ) r_{\mathrm{per}}(z) ]^k, \end{aligned}$$

(5.10)

we see that $N(\psi )$ can be written as the formal series

$$\begin{aligned} N(\psi ) = \sum _{k=2}^{\infty } {\text {den}}[N_k(\psi )], \end{aligned}$$

(5.11)

where

$$\begin{aligned} N_k(\psi ) := \oint r_{\mathrm{per}}(z) [(-\psi ) r_{\mathrm{per}}(z) ]^k. \end{aligned}$$

(5.12)

Proposition 5.3

Let Assumption [A1] hold and let $N_2$ be given by (5.12). Assume that $\Vert \nabla \psi \Vert _{L^2} = o(1)$, then we have the estimate

$$\begin{aligned}&\Vert {\text {den}}[N_k(\psi )]\Vert _{L^2} \lesssim \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3} \Vert \psi \Vert _{H^j}^{k-2},\quad j = 0,1, \end{aligned}$$

(5.13)

where the constants associated with $\lesssim$ are independent of $\beta$.

Fig. 2

The deformation of the contour $\Gamma$ is indicated by the blue dashed line. The spectrum of $h_{\mathrm{per}}$ is denoted by solid black line. The poles of $f_{T}(z-\mu )$ are denoted by the black dots. The number c denotes the lower bound $h_{\mathrm{per}} > -c + 1$

Proof

Below, we use the notation $r = r_{\mathrm{per}}(z)$, where $r_{\mathrm{per}}(z)$ is given in (3.8), and the estimate (see (4.32))

$$\begin{aligned} \Vert (1-\Delta )^{\alpha } r\Vert \lesssim \&d^{\alpha -1}\lesssim 1, \end{aligned}$$

(5.14)

for $\alpha \in [0, 1]$ and $z\in \Gamma$, where

$$\begin{aligned} d \equiv d(z):=\mathrm {dist}(z, \sigma (h_{\mathrm{per}}))\ge \frac{1}{4} . \end{aligned}$$

(5.15)

We use the $L^2$–$L^2$ duality to estimate the $L^2$ norm of ${\text {den}}[N_k(\psi )]$. We have, by (2.5) and definition (5.12),

$$\begin{aligned} \Vert {\text {den}}[N_k(\psi )]\Vert _{L^2}&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \int f {\text {den}}[N_k(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \mathrm {Tr}[f N_k(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \oint \mathrm {Tr}(f r (\psi r)^k)\right| . \end{aligned}$$

(5.16)

(In the last two lines, f is considered as a multiplication operator.)

Let $f \in L^2$ and recall the Schatten norm $\Vert \cdot \Vert _{S^p}$ defined in (2.3). Using the non-abelian Hölder’s inequality $1 = \frac{1}{2} + \frac{1}{6} + \frac{1}{3}+ \frac{1}{\infty }$, we see that, for $k\ge 2$,

$$\begin{aligned} |\mathrm {Tr}( f r (\psi r)^k)| \lesssim&\Vert f r\Vert _{S^2} \Vert \psi r\Vert _{S^6} \Vert \psi r\Vert _{S^3}\Vert \psi r\Vert ^{k-2} \, . \end{aligned}$$

(5.17)

Next, we use the operator trace-class estimate $\Vert A\Vert _{S^3}^3 = \mathrm {Tr}(|A|^3) \le \Vert A\Vert \mathrm {Tr}(|A|^2) = \Vert A\Vert \Vert A\Vert _{S^2}^2\le \Vert A\Vert _{S^6} \Vert A\Vert _{S^2}^2$ to obtain

$$\begin{aligned} \Vert A\Vert _{S^3}\le \Vert A\Vert _{S^6}^{1/3} \Vert A\Vert _{S^2}^{2/3}. \end{aligned}$$

(5.18)

Using this equality to estimate the third factor in (5.17) and the standard relative bounds $\Vert \psi r\Vert \lesssim \Vert \psi \Vert _{L^2}$ and $\Vert \psi r\Vert \lesssim \Vert \psi \Vert _{L^6}\lesssim \Vert \psi \Vert _{\dot{H}^1}$, we bound the r.h.s. of (5.17) as

$$\begin{aligned} |\mathrm {Tr}( f r (\psi r)^k)| \lesssim&\Vert f r\Vert _{S^2} \Vert \psi r\Vert _{S^6}^{4/3} \Vert \psi r\Vert _{S^2}^{2/3}\Vert \psi \Vert _{\dot{H}^i}^{k-2},\ i=0, 1. \end{aligned}$$

(5.19)

For a typical term on the r.h.s., we have $\Vert g r\Vert _{S^p}\le \Vert g (1-\Delta )^{-\alpha _p} \Vert _{S^p} \Vert (1-\Delta )^{\alpha _p} r\Vert$, with $3/(2p)<\alpha _p<1, p>3/2$, which, together with Kato–Seiler–Simon’s inequality (2.14) and inequality (5.14), gives

$$\begin{aligned} \Vert g r\Vert _{S^p}\lesssim \Vert g\Vert _{L^p}d^{\alpha _p-1},\ 3/(2p)<\alpha _p<1, p>3/2. \end{aligned}$$

Applying this estimate to each of the first three factors on the r.h.s. of (5.19) and using the Gagliardo–Nirenberg–Sobolev inequality (2.18), we find

$$\begin{aligned} \big |\mathrm {Tr}( f r (\psi r)^k)\big |\lesssim&d^{-4/3}\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}\Vert \psi \Vert _{\dot{H}^j}^{k-2}, \end{aligned}$$

(5.20)

for $j=0, 1$. Recalling definition (5.15) of $d \equiv d(z)$, we see that the integral on the r.h.s. of (5.16) converges absolutely. Equations (5.20), (5.15), (3.7) and (3.6) give

$$\begin{aligned} \left| \oint \mathrm {Tr}( f r (\psi r)^k) \right| \lesssim&\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}\Vert \psi \Vert _{\dot{H}^j}^{k-2}, \end{aligned}$$

(5.21)

for $j=0, 1$. Equations (5.16) and (5.21) imply (5.13). $\square$

Now, we complete the proof of Proposition 5.2. Proposition 5.3 shows that if $\Vert \psi \Vert _{L^2} < \infty$ and either $\Vert \psi \Vert _{L^2} = o(1)$ or $\Vert \psi \Vert _{\dot{H}^{1}} = o(1)$, then series (5.11) converges absolutely in $L^2$.

Now, using series (5.11), we write

$$\begin{aligned} N(\psi _1) - N(\psi _2)= \sum _{k \ge 2} {\text {den}}[N_k(\psi _1) - N_k(\psi _2)] \, . \end{aligned}$$

(5.22)

By definition (5.12), $N_k(\psi )$ is an k-th degree monomial in $\phi$. Hence, we can expand $N_k(\psi _1) - N_k(\psi _2)$ in the following telescoping form

$$\begin{aligned} x^k&- y^k = x^{k-1}(x-y) + x^{k-2}(x-y)y + \cdots + (x-y)y^k\, . \end{aligned}$$

(5.23)

The proof of Proposition 5.2 follows by applying appropriate and straightforward extension of Proposition 5.3 to each term in the expansion of $N_k(\psi _1) - N_k(\psi _2)$ given in (5.23). $\square$

Appendices

Appendix A: $\epsilon (T) \rightarrow \epsilon (0)$ as $T \rightarrow 0$

Lemma A 1

Let $\text {xc}= 0$. Then $\epsilon \equiv \epsilon (T) \rightarrow \epsilon (0)$ as $T \rightarrow 0$, where $\epsilon (0)$ is the dielectric constant for $T=0$ obtained in [7].

Proof

We see from (1.35) below that $\epsilon (T), T=1/\beta ,$ is of the form

$$\begin{aligned} \epsilon (T) = \frac{1}{2\pi i} \int _\Gamma f_T(z-\mu ) X(z) \end{aligned}$$

(A.1)

where X(z) is some holomorphic function on $\mathbb {C}\backslash \mathbb {R}$, independent of $\beta$, and remains holomorphic on the real axis where the gap of $h_{\mathrm{per}}$ occurs. On $\mathbb {R}$, we note that $f_{\mathrm{FD}}(\beta x)$ converges to the indicator function $\chi _{(-\infty , 0)}$ as $\beta \rightarrow \infty$. If we take $\beta \rightarrow \infty$, the integral

$$\begin{aligned} \frac{1}{2\pi i} \int _\Gamma f_T(z-\mu ) X(z) \end{aligned}$$

(A.2)

converges to $\frac{1}{2\pi i} \int _{G_1} X(z)$ where $G_1$ is any contour around the part of the spectrum of $h_{\mathrm{per}}$ that is less than $\mu _{\mathrm{per}}$. This is the same expression as in [7] after inserting $1 = \sum _i |\varphi _i \rangle \langle \varphi _i|$ for each resolvent of $h_{\mathrm{per}}$ in X(z) where the $\varphi _i$’s are eigenvectors of $h_{\mathrm{per}}$. $\square$

Appendix B: Bounds on m and V

In this section, we prove bounds on m and V given (1.33) and (4.28). Note that $m=\Vert V\Vert _{L^1_{\mathrm{per}}}$. Since $f'_{T} < 0$ ($T= 1/\beta$), (4.28) implies that $V > 0$ and therefore, by (4.28), $\Vert V\Vert _{L^1_{\mathrm{per}}} = \int _\Omega V$, where $\Omega$ is a fundamental domain of $\mathcal {L}$ (see Sect. 1.5), which yields

$$\begin{aligned} m=\int _\Omega V = - \mathrm {Tr}_{L^2_{\mathrm{per}}} f'_{T}(h_{\mathrm{per}, 0}-\mu ),\ \qquad \mu =\mu _{\mathrm{per}}. \end{aligned}$$

(B.1)

Lemma B.1

Let Assumption [A1] hold and $\eta _0$ be given in (1.27). Then

$$\begin{aligned} m=\Vert V\Vert _{L^1_{\mathrm{per}}} \ge \frac{1}{4} \beta e^{-\beta \eta _0}, \end{aligned}$$

(B.2)

where $\eta _0$ is given in (1.27).

Proof

Using that $\eta _0$ is the smallest distance between $\mu = \mu _{\mathrm{per}}$ and the spectrum of $h_{\mathrm{per}, 0}$ (see (1.27)) and Eq. (B.1) and replacing $\mathrm {Tr}_{L^2_{\mathrm{per}}} f'_{T}(h_{\mathrm{per}, 0}-\mu )$ by the contribution of the eigenvalue of $h_{\mathrm{per}, 0}$ closest to $\mu$, we find

$$\begin{aligned} m \ge&- f'_{T}(\eta _0) = \beta \frac{e^{\beta \eta _0}}{(1+e^{\beta \eta _0})^2} \ge \frac{1}{4} \beta e^{-\beta \eta _0}. \end{aligned}$$

(B.3)

This gives (B.2). $\square$

Lemma B.2

Let Assumption [A1] hold. Then, for $1 \le p \le \infty$,

$$\begin{aligned} \Vert V\Vert _{L^p_{\mathrm{per}}} \lesssim \beta e^{-\eta _0\beta }. \end{aligned}$$

(B.4)

Proof

We do the case for $p=1$ and $p=\infty$, and conclude the lemma by interpolation. By Assumption [A1], the potential $\phi _{\mathrm{per}}$ is bounded. Thus, $h_{\mathrm{per}, 0}$ has only discrete spectrum on $L^2_{\mathrm{per}}(\mathbb {R}^3)$ and

$$\begin{aligned} \frac{1}{\beta }\Vert V\Vert _{L^1_{\mathrm{per}}}&= \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})} \frac{ e^{\beta (\lambda -\mu )}}{(1+e^{\beta (\lambda -\mu )})^2} \end{aligned}$$

(B.5)

$$\begin{aligned}&\le \sum _{\mu > \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta (\lambda -\mu )}+ \sum _{\mu < \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{-\beta (\lambda -\mu )} \end{aligned}$$

(B.6)

$$\begin{aligned}&= \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta |\lambda -\mu |} . \end{aligned}$$

(B.7)

Again, we use that $\eta _0$ is the smallest distance between $\mu = \mu _{\mathrm{per}}$ and the spectrum of $h_{\mathrm{per}, 0}$ (see (1.27)). Peeling the eigenvalue(s) closest to $\mu$ and letting $\eta _0+\xi$ stand for the distance between $\mu$ and the rest of the spectrum $\sigma (h_{\mathrm{per}, 0})$, we find, for some constant c,

$$\begin{aligned} \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0})}&e^{\beta |\lambda -\mu |} =ce^{\beta \eta _0}+ \sum _{\lambda \in \sigma (h_{\mathrm{per}, 0}), |\lambda -\mu |\ge \eta _0+\xi }e^{\beta |\lambda -\mu |}. \end{aligned}$$

(B.8)

We estimate the sum on the r.h.s. by an integral as follows. Since the potential $\phi _{\mathrm{per}}$ is infinitesimally bounded with respect to $-\Delta$, the eigenvalues of $h_{\mathrm{per}, 0}$ go to infinity at a similar rate as those of $-\Delta$ (on $L^2_{\mathrm{per}}(\mathbb {R}^3)$), i.e. as $n^2$. Thus, assuming that for $\lambda$ sufficiently large, the nth eigenvalue $\lambda _n\approx n^2$ has the degeneracy of the order $O(n^k), k\ge 0$, we conclude that

$$\begin{aligned}&\sum _{\mu < \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{-\beta (\lambda -\mu )}\lesssim \int _{x^2 \ge \mu +\eta _0+\xi } x^k e^{-\beta (x^2 -\mu )} d x \nonumber \\&\quad =\frac{1}{2} \int _{y\ge \eta _0+\xi } (y+\mu )^{\frac{k-1}{2}} e^{-\beta y} d y \lesssim \frac{1}{\beta }\mu ^{\frac{k-1}{2}} e^{-\beta (\eta _0+\xi )}. \end{aligned}$$

(B.9)

For the first sum in (B.6), we consider separately the cases $\mu \lesssim 1$ and $\mu \gg 1$ and, in the 2nd case, break the sum into the sums over $\lambda \lesssim 1$ and $\lambda \gg 1$. In the first three situations, the estimate is straightforward and in the last one, we proceed as in (B.9) to obtain

$$\begin{aligned} \sum _{\mu -\eta _0-\xi \ge \lambda \in \sigma (h_{\mathrm{per}, 0})}e^{\beta (\lambda -\mu )}\lesssim \frac{1}{\beta }\mu ^{\frac{k-1}{2}} e^{-\beta \eta _0}. \end{aligned}$$

This proves the lemma for $p=1$.

Let $W^{4,1}_{\mathrm{per}}$ be the usual Sobolev space associated to $L^1_{\mathrm{per}}$ involving up to 4 derivatives. For the case $p=\infty$, we use the Sobolev inequality

$$\begin{aligned} \Vert f\Vert _\infty \lesssim \Vert f\Vert _{W^{4,1}_{\mathrm{per}}} \end{aligned}$$

(B.10)

for $f \in W^{4,1}_{\mathrm{per}}$. Thus, it suffices for us to estimate $\Vert \nabla ^j V\Vert _{L^1_{\mathrm{per}}}, j=0,\ldots ,4$. To this end, we note that

$$\begin{aligned} \nabla {\text {den}}(A) = {\text {den}}( [\nabla , A] ) \end{aligned}$$

(B.11)

for an operator A on $L^2_{\mathrm{per}}(\mathbb {R}^3)$. Thus, it suffices that we estimate the trace 1-norm of $\nabla ^s f_{T}'(h_{\mathrm{per}, 0}-\mu ) \nabla ^{4-s}$ on $L^2_{\mathrm{per}}(\mathbb {R}^3)$ for $s=0,\ldots ,4$. Since the potential $\phi _{\mathrm{per}}$ is bounded together with all its derivatives, we have, for $s=0, \dots , 4$,

$$\begin{aligned} \Vert \nabla ^s h^{-s/2}\Vert \lesssim 1, \end{aligned}$$

(B.12)

where $h:=h_{\mathrm{per}, 0}+c$, with $c>0$ s.t. $h_{\mathrm{per}, 0}+c>0$. Indeed, to fix ideas, consider one of the terms, say, $\Vert \nabla ^3 h^{-3/2}\Vert$. We have $\Vert \nabla ^3f\Vert ^2\le \Vert (-\Delta )^{3/2}f\Vert ^2=\langle f, (h+\phi _{\mathrm{per}}-c)^3f\rangle$. Taking $f = h^{-3/2}u$, expanding the binomial $(h+\phi _{\mathrm{per}}-c)^3$ and commuting the operator h in the resulting terms $h^2\phi _{\mathrm{per}}$ and $\phi _{\mathrm{per}} h^2$ to the right and left, respectively, and estimating the resulting commutators, $[h, \phi ]$ and $[\phi _{\mathrm{per}}, h]=-[h, \phi _{\mathrm{per}}]$, we arrive at the estimate $\Vert \nabla ^3 h^{-3/2}\Vert \lesssim 1$ as claimed. (B.12) implies also that $\Vert h^{-2+s/2}\nabla ^{4-s}\Vert \lesssim 1$, for $j=0, \dots , 4$. As the result, we have

$$\begin{aligned} \Vert \nabla ^s f_{T}'(h_{\mathrm{per}, 0}-\mu ) \nabla ^{4-s}\Vert _{S^1}\lesssim \Vert g(h_{\mathrm{per}, 0})\Vert _{S^1}, \end{aligned}$$

where $g(x):=-(x+c)^{s/2} f_{T}'(x-\mu ) (x+c)^{2-s/2}\ge 0$. Hence, it suffices to estimate $\Vert g(h_{\mathrm{per}, 0})\Vert _{S^1}=\mathrm {Tr}[g(h_{\mathrm{per}, 0})]$. The latter can be done the same way as the case for $p=1$ by summing eigenvalues of $h_{\mathrm{per}, 0}$ and the lemma is proved. $\square$

Appendix C: Bound on $M_\delta$

In an analogy to $L_{\mathrm{per}}^2\equiv L_{\mathrm{per}}^2(\mathbb {R}^3)$ given in (1.19), we let

$$\begin{aligned} L^2_{\mathrm{per}, \delta }\equiv L^2_{\mathrm{per}, \delta }(\mathbb {R}^3) := \{ f \in L^2_{\mathrm{loc}}(\mathbb {R}^3) : f \text { is } \mathcal {L}_\delta \text {-periodic } \}. \end{aligned}$$

(C. 1)

Moreover, we recall $\nabla ^{-1} := \nabla (-\Delta )^{-1}$ (see (3.56)). The main result of this appendix is the following

Proposition C.1

Let Assumption [A1] hold. Then the operator $M_\delta$ can be decomposed as

$$\begin{aligned} M_\delta = M_\delta ' + M_\delta '' \, , \end{aligned}$$

(C. 2)

with the operator $M_\delta '$ and $M_\delta ''$ satisfying the estimates

$$\begin{aligned}&\Vert {\bar{P_r}} \nabla ^{-1} M_\delta ' P_r\varphi \Vert _{L^2} \lesssim \delta ^{-1} \Vert V\Vert _{L^2_{\mathrm{per}}} \Vert P_r\varphi \Vert _{L^2}, \end{aligned}$$

(C. 3)

$$\begin{aligned}&\Vert {\bar{P_r}} \nabla ^{-1} M_\delta '' P_r\nabla ^{-1} \varphi \Vert _{L^2} \lesssim \Vert P_r\varphi \Vert _{L^2}. \end{aligned}$$

(C. 4)

Proof of Proposition C.1

Proposition 4.2 and the rescaling relation (3.34) imply the explicit form for the k-fibers of $M_\delta$:

Lemma C.2

Then $M_\delta$ has a Bloch–Floquet decomposition (2.34) with $\mathcal {L}= \mathcal {L}_\delta$, whose $k-$fiber $M_{\delta , k}$ acting on $L^2_{\mathrm{per}, \delta }$ is given by

$$\begin{aligned} M_{\delta ,k} f =&- \delta {\text {den}}\left[ \oint r^\delta _{\mathrm{per, 0}}(z)f r^\delta _{\mathrm{per}, k}(z)\right] \end{aligned}$$

(C. 5)

where $f \in L^2_{\mathrm{per}, \delta }$ and, on $L^2_{\mathrm{per}, \delta }$,

$$\begin{aligned}&r^\delta _{\mathrm{per}, k}= (z- h^\delta _{\mathrm{per}, k})^{-1}, \quad h^\delta _{\mathrm{per}, k}= \delta ^2 (-i\nabla -k)^2 + \delta \phi _{\mathrm{per}}^\delta . \end{aligned}$$

(C. 6)

We decompose the operator $M_{\delta , k}$ acting on $L^2_{\mathrm{per}, \delta }$ as

$$\begin{aligned} M_{\delta , k}&=: M_{\delta , 0} + M_{\delta , k}'' \, , \end{aligned}$$

(C. 7)

where $M_{\delta , 0} = M_{\delta , k=0}$ and $M_{\delta , k}'$ is defined by the expression (C. 7). We define operators $M_\delta '$ and $M_\delta ''$ on $L^2(\mathbb {R}^2)$ via

$$\begin{aligned} M_\delta '&:= \int _{\Omega _\delta ^*}^\oplus d{\hat{k}} \, M_{\delta , 0} \varphi , \end{aligned}$$

(C. 8)

$$\begin{aligned} M_\delta ''&:= \int _{\Omega _\delta ^*}^\oplus d{\hat{k}} \, M''_{\delta , k} \varphi \, , \end{aligned}$$

(C. 9)

where $\Omega _\delta ^*$ is a fundamental cell of the reciprocal lattice to $\mathcal {L}_\delta$ and $d{\hat{k}} = |\Omega _\delta ^*|^{-1} dk$. By Lemma C.2 and definition (C. 7), the latter operators satisfy (C. 2).

Lemma C.3

$M_\delta '$ (see (C. 8)) restricted to the range of $P_r$ is a multiplication operator given by

$$\begin{aligned} (M_\delta ' P_r\varphi )(x) = V_\delta (x) (P_r\varphi )(x), \end{aligned}$$

(C. 10)

where

$$\begin{aligned} V_\delta (x) = -\delta ^{-2} {\text {den}}\left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] (\delta ^{-1}x) \, , \end{aligned}$$

(C. 11)

with $h_{\mathrm{per}, 0}$ given in (4.6) (with $k = 0$).

Proof

By (C. 5) and definition of $M_\delta '$ in (C. 8), we see that

$$\begin{aligned} M_\delta '&P_r \varphi = - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, \delta {\text {den}}\left[ \oint r^\delta _{\mathrm{per, 0}}(z)(P_r \varphi )_k r^\delta _{\mathrm{per, 0}}(z)\right] . \end{aligned}$$

(C. 12)

By Corollary 2.4 and the Cauchy integral formula,

$$\begin{aligned} M_\delta ' P_r\varphi&= - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, \delta {\text {den}}\left[ \oint (r^\delta _{\mathrm{per, 0}}(z))^2 \right] |\Omega _\delta |^{-1} {\hat{\varphi }}(k) \end{aligned}$$

(C. 13)

$$\begin{aligned}&= - \int ^\oplus _{\Omega _\delta ^*} d{\hat{k}} \, {\text {den}}[f_{T}'(h^\delta _{\mathrm{per}, 0} - \mu )] |\Omega _\delta |^{-1} {\hat{\varphi }}(k) . \end{aligned}$$

(C. 14)

where $r^\delta _{\mathrm{per, 0}}(z)$ and $h^\delta _{\mathrm{per}, 0}$ are given in (C. 6). Applying the inverse Bloch–Floquet transform (2.24), (C. 14) implies

$$\begin{aligned} M_\delta '&P_r\varphi = -\delta {\text {den}}[f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) ] \int _{\Omega _\delta ^*} d{\hat{k}} \, e^{-ikx} |\Omega _\delta |^{-1}{\hat{\varphi }}(k). \end{aligned}$$

(C. 15)

Since $d{\hat{k}}$ is normalized by the volume $|\Omega _\delta ^*|$ (which is independent of the choice of the cell), (C. 15) shows

$$\begin{aligned} M_\delta ' P_r\varphi =&- \delta {\text {den}}\left[ f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) \right] P_r\varphi . \end{aligned}$$

(C. 16)

By Lemma 2.7 and recalling the definition of $U_{\delta }$ from (2.45), we see that

$$\begin{aligned} \delta {\text {den}}\left[ f_{T}'(h^\delta _{\mathrm{per}, 0}-\mu ) \right]&= \delta {\text {den}}\left[ U_{\delta }f_{T}'(h_{\mathrm{per}, 0}-\mu ) U_{\delta }^* \right] \end{aligned}$$

(C. 17)

$$\begin{aligned}&= \delta ^{-2} {\text {den}}\left[ f_{T}'(h_{\mathrm{per}, 0}-\mu ) \right] (\delta ^{-1}x) \, , \end{aligned}$$

(C. 18)

where $h_{\mathrm{per}, 0} = h_{\mathrm{per}, 0}^{\delta = 1}$, which together with (C. 16) gives (C. 10)–(C. 11). $\square$

Proof of (C. 3)

Let $V_\delta$ be given in (C. 11). Since the Bloch–Floquet decomposition is unitary, we see, by Lemma C.3 and Corollary 2.4, that

$$\begin{aligned} \Vert M_\delta ' P_r\varphi \Vert _{L^2}^2 =&\Vert V_\delta P_r\varphi \Vert _{L^2}^2 = \int _{\Omega _\delta ^*} d{\hat{k}} \Vert V_\delta |{\hat{\varphi }}(k)| |\Omega _\delta |^{-1} \Vert _{L^2_{\mathrm{per}, \delta }}^2, \end{aligned}$$

(C. 19)

where $L^2_{\mathrm{per}, \delta }$ is given in (C. 1). Using the fact that $d{\hat{k}} = |\Omega _\delta ^*|^{-1} dk$ and $|\Omega _\delta | = \delta ^3 |\Omega |$, (C. 19) implies

$$\begin{aligned} \Vert M_\delta ' P_r\Vert _{L^2}^2 =&\delta ^{-3} |\Omega |\Vert V_\delta \Vert _{L^2_{\mathrm{per}, \delta }}^2 \Vert P_r\varphi \Vert _{L^2}^2. \end{aligned}$$

(C. 20)

By a change of variable, we see that $\Vert V_\delta \Vert _{L^2_{\mathrm{per}, \delta }} = \delta ^{-1/2}\Vert V \Vert _{L^2_{\mathrm{per}}},$ where V is given by (4.28). Combing with (C. 20), the fact ${\bar{P_r}} (-i\nabla )^{-1} \lesssim r^{-1}$ (where $\nabla ^{-1}$ is given in (3.56)) and $r^{-1}= a^{-1}\delta \lesssim \delta$ (see (3.38)) yields Eq. (C. 3). $\square$

Proof of (C. 4)

Let $M_\delta ''$ be given by (C. 9) and $k^{-1} := k/|k|^2$. Let $\varphi \in L^2(\mathbb {R}^3)$. By Corollary 2.4, we have

$$\begin{aligned} (P_r\nabla ^{-1} \varphi )_k= k^{-1} {\hat{\varphi }}(k) |\Omega _\delta |^{-1} \chi _{B(r)}(k). \end{aligned}$$

(C. 21)

This gives $M_\delta '' P_r\nabla ^{-1} \varphi = |\Omega _\delta |^{-1}\int ^\oplus _{B(r)} d{\hat{k}} M''_{\delta , k} k^{-1} {\hat{\varphi }}(k)$. Since the Bloch–Floquet decomposition is unitary, we see, using (C. 21), that

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1}&\varphi \Vert _{L^2}^2 \nonumber \\&= |\Omega _\delta |^{-2}\int _{B_r} d{\hat{k}} \, \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}^2 |k|^{-2} |{\hat{\varphi }}(k)|^2. \end{aligned}$$

(C. 22)

Since $d {\hat{k}} = |\Omega ^*|^{-1} dk = |\Omega | dk$ and $|\Omega _\delta | = \delta ^3|\Omega |$, (C. 22) is bounded as

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1}&\varphi \Vert _{L^2}^2 \nonumber \\&\lesssim \delta ^{-3} \sup _{k \in B_r} \left( \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}^2 |k|^{-2} \right) \Vert P_r\varphi \Vert _{L^2}^2, \end{aligned}$$

(C. 23)

where $1 \in L^2_{\mathrm{per}, \delta }$ is the constant function 1 and $L^2_{\mathrm{per}, \delta }$ is given in (C. 1).

By (C. 5) and (C. 7), we have, for $M''_{\delta , k}$ given in (C. 7), that

$$\begin{aligned} M''_{\delta , k} \varphi =&- \delta {\text {den}}\left[ \oint r_{\mathrm{per}, 0}(z) \varphi (r^\delta _{\mathrm{per}, k}(z)-r^\delta _{\mathrm{per}, 0}(z)) \right] . \end{aligned}$$

(C. 24)

Since $r^\delta _{\mathrm{per}, k}(z)-r_{\mathrm{per}, 0}(z)=r^\delta _{\mathrm{per}, 0}(z)A_k r^\delta _{\mathrm{per}, k}(z)$, where $A_k:=-2 (-i\nabla ) \delta k+\delta ^2 |k|^2$, this gives

$$\begin{aligned} M''_{\delta , k} \varphi =&- \delta {\text {den}}\oint \left[ r^\delta _{\mathrm{per}}(z) \varphi r^\delta _{\mathrm{per}}(z) A_k r^\delta _{\mathrm{per}, k}(z) \right] . \end{aligned}$$

(C. 25)

By the rescaling relation (3.34) and (C. 25), we see that

$$\begin{aligned} \Vert M''_{\delta , k} 1\Vert _{L^2_{\mathrm{per}, \delta }}&= \Vert U_\delta ^* M''_{\delta , k}U_\delta \cdot U_\delta ^* 1\Vert _{L^2_{\mathrm{per}}} = \delta ^{3/2-2}\Vert M''_{1, k} 1\Vert _{L^2_{\mathrm{per}}}\nonumber \\&= \delta ^{-1/2} \big \Vert {\text {den}}\big [ \oint r_{\mathrm{per}}^2(z) A_kr_{\mathrm{per}, \delta k}(z) \big ] \big \Vert _{L^2_{\mathrm{per}}}. \end{aligned}$$

(C. 26)

By (C. 26), notation $A_k:=-2 (-i\nabla ) \delta k+\delta ^2 |k|^2$ and inequality (4.32), with $\alpha =0,1/2$, we obtain, for $|k| \le r$,

$$\begin{aligned} \Vert M''_{\delta , k}1\Vert _{L^2_{\mathrm{per}}} |k|^{-1}&\lesssim \delta ^{-1/2+1} + \delta ^{-1/2+2} r \nonumber \\&= \delta ^{1/2} + \delta ^{3/2} r \, . \end{aligned}$$

(C. 27)

By (3.38) and (C. 23), Eq. (C. 27) shows that

$$\begin{aligned} \Vert M_\delta '' P_r\nabla ^{-1} \varphi \Vert _{L^2} \lesssim \delta ^{-1} . \end{aligned}$$

(C. 28)

This bound, the observation that $\Vert {\bar{P_r}} \nabla ^{-1}\Vert _\infty \lesssim r^{-1}$ (see (3.37)) and the definition $r =a/\delta > rsim 1/\delta$ imply Eq. (C. 4). $\square$

This completes the proof of Proposition C.1. $\square$

We use Proposition C.1 to prove the following

Proposition C.4

Let Assumption [A1] hold and let $\beta e^{-\eta _0\beta }\lesssim 1$ (which is weaker than Assumption [A3]). Then the operator $M_\delta$ is bounded as

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}}&M_\delta f\Vert _{\dot{H}^1} \lesssim \Vert f\Vert _{\delta }. \end{aligned}$$

(C. 29)

Proof of Proposition C.4

Decomposing $M_\delta$ according to (C. 2) and using bounds Eqs. (C. 3) and (C. 4) of Proposition C.1, we see that

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}} M_\delta P_r&\varphi \Vert _{L^2} \le \Vert \nabla ^{-1}{\bar{P_r}} M_\delta ' P_r\varphi \Vert _{L^2} + \Vert \nabla ^{-1}{\bar{P_r}} M_\delta '' P_r\varphi \Vert _{L^2}\nonumber \\&\lesssim \delta ^{-1} \Vert V \Vert _{L^2_{\mathrm{per}}} \Vert f\Vert _{L^2} + \Vert \nabla f\Vert _{L^2}, \end{aligned}$$

(C. 30)

where V is given in (4.28). Using $\Vert V\Vert _{L^2_{\mathrm{per}}} \le \Vert V\Vert _{L^\infty }^{1/2} \Vert V\Vert _{L_{\mathrm{per}}^1}^{1/2}$ and the definition $m:=\Vert V\Vert _{L^1_{\mathrm{per}}}$ in (C. 30) gives

$$\begin{aligned} \Vert \nabla ^{-1}{\bar{P_r}}&M_\delta f\Vert _{\dot{H}^1} \lesssim \Vert V \Vert _{L^\infty }^{1/2} (\delta ^{-1} m^{1/2}\Vert f\Vert _{L^2}) + \Vert \nabla f\Vert _{L^2} \, . \end{aligned}$$

(C. 31)

Lemma B.2 and definition (3.45) imply (C. 29). $\square$

Appendix D: Refined Nonlinear Estimates

Let $N_\delta$ be given implicitly by (3.31) and recall the definition of the $B_{s,\delta }$ norm from (3.45). Let $\dot{H}^{0}\equiv L^2$. In this section we prove estimates on $N_\delta$.

Proposition D.1

Let Assumption [A1] hold. If $\Vert \varphi _1\Vert _{B_{s,\delta }}, \Vert \varphi _2\Vert _{B_{s,\delta }} = o(\delta ^{-1/2})$, then we have the estimate

$$\begin{aligned} \Vert N_\delta&(\varphi _1) - N_\delta (\varphi _2)\Vert _{L^2} \nonumber \\&\lesssim e^{-\beta }m^{ -1/3} \delta ^{-1/2}(\Vert \varphi _1\Vert _{B_{s,\delta }} + \Vert \varphi _2\Vert _{B_{s, \delta }})\Vert \varphi _1 - \varphi _2\Vert _{\dot{H}^1} . \end{aligned}$$

(D.1)

We derive Proposition D.1 from its version with $\delta = 1$ by rescaling. For $\delta = 1$, we have the following result.

Proposition D.2

Let Assumption [A1] hold $\Vert \psi \Vert _{L^2} = o(1)$. Then $N:=N_{\delta =1}$ satisfies the estimate

$$\begin{aligned} \Vert N(\psi _1) - N(\psi _2)\Vert _{L^2}&\lesssim \sum _{j=1}^2\bigg [ (\Vert \psi _j\Vert _{\dot{H}^1})\Vert \psi _1 - \psi _2\Vert _{\dot{H}^1} \nonumber \\&\quad + e^{-\beta } \big (\Vert \psi _j\Vert _{\dot{H}^1}^{1/3} \Vert \psi _j\Vert _{L^2}^{2/3} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}} \nonumber \\&\quad + \Vert \psi _j\Vert _{\dot{H}^{1}} \Vert \psi _1 - \psi _2\Vert _{\dot{H}^{1}}^{1/3}\Vert \psi _1 - \psi _2\Vert _{L^2}^{2/3}\big )\bigg ]. \end{aligned}$$

(D.2)

The derivation of Proposition D.1 from Proposition D.2 is same as that of Proposition 5.1 from Proposition 5.2 and we omit it here.

Proof of Proposition D.2

Let $h_{\mathrm{per}}$ and $r_{\mathrm{per}}(z)$ be given in (3.8). We use the relations (5.6)–(5.12) in the proof of Proposition 5.2. Following the latter proof we see that it suffices to improve the estimate of $N_k(\psi )$ in Proposition 5.3, to which we proceed. $\square$

Proposition D.3

Let Assumption [A1] hold and let $N_k$ be given by (5.12). Assume that $\Vert \nabla \psi \Vert _{L^2} = o(1)$, then, for any $k\ge 2$, we have the estimate

$$\begin{aligned}&\Vert {\text {den}}[N_k(\psi )]\Vert _{L^2} \lesssim \Vert \nabla \psi \Vert _{L^2}^k + e^{-\beta } \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}\delta _{k, 2}, \end{aligned}$$

(D.3)

where the constants associated with $\lesssim$ are independent of $\beta$ and $\delta _{k, 2}$ is the Kronecker delta.

Proof

We begin with $k=2$. To improve upon estimate (5.13), we, following [8], use the partition of unity

$$\begin{aligned} P_1+ P_2= \mathbf {1}, \text { with } P_1 := \chi _{h_{\mathrm{per}} < \mu } \text { and } P_2 := \chi _{h_{\mathrm{per}} \ge \mu }. \end{aligned}$$

(D.4)

Let $R_i \equiv R_i (z)= r_{\mathrm{per}}(z) P_i$ where $i=1, 2,$$P_i$ and $r_{\mathrm{per}}(z)$ are given in (D.4) and (3.8). Recalling definition (5.12) of $N_k(\psi )$ and inserting the partition of unity, $P_1 + P_2 = \mathbf {1}$, after each R in the integrand of (5.12), we arrive at

$$\begin{aligned} N_2(\psi ) = \sum _{a,b,c = 1,2}N^{abc}_2(\psi ) \, , \end{aligned}$$

(D.5)

where the $N_2^{abc}(\psi )$, for $a,b,c =1, 2$, denote the operators

$$\begin{aligned} N^{abc}_2(\psi ) =&\oint R_a \phi R_b \psi R_c \, . \end{aligned}$$

(D.6)

We estimate the terms individually. Below, we use the estimate (see (4.32))

$$\begin{aligned} \Vert (1-\Delta )^{\alpha } R_i(z)\Vert \lesssim \&d^{\alpha -1}\lesssim 1,\ i=1, 2, \end{aligned}$$

(D.7)

for $\alpha \in [0, 1]$ and $z\in \Gamma$, where

$$\begin{aligned} d \equiv d(z):=\mathrm {dist}(z, \sigma (h_{\mathrm{per}}))\ge \frac{1}{4} . \end{aligned}$$

(D.8)

Case 1 (121) and (212). We estimate the case for (121), the other case is done similarly. Since $P_1P_2 = 0$, we write

$$\begin{aligned} N^{(121)}_2(\psi )&= \oint R_1 \psi R_2 P_2 \psi R_1 \end{aligned}$$

(D.9)

$$\begin{aligned}&= \oint R_1 [P_1, \psi ]P_2 R_2 P_2 [\psi , P_1] R_1 \, . \end{aligned}$$

(D.10)

Applying Lemma 2.1 and Eq. (D.7) to the r.h.s. and using that the operator norm is bounded by the $I^2$ norm, we find

$$\begin{aligned} \Vert {\text {den}}[N^{(121)}_2(\psi )] \Vert _{L^2}&\lesssim \Vert (1-\Delta )^{3/4+\epsilon } N^{(121)}_2(\phi )\Vert _{S^2} \end{aligned}$$

(D.11)

$$\begin{aligned}&\lesssim \left| \oint \right| d^{-3} \Vert [P_1, \psi ]P_2 \Vert _{S^2}^2. \end{aligned}$$

(D.12)

where, recall,

$$\begin{aligned} \left| \oint \right| :=&\frac{1}{2\pi } \int _\Gamma dz |f_{T} (z-\mu )|. \end{aligned}$$

(D.13)

A key observation allowing us to obtain an improved estimate is that the commutators lead to gradient estimates:

Lemma D.4

Let Assumption [A1] hold, we have the estimate

$$\begin{aligned} \Vert [P_i, \psi ] \Vert _{S^2} \lesssim \Vert \nabla \psi \Vert _{L^2},\ i=1, 2 . \end{aligned}$$

(D.14)

Proof of Lemma D.4

Since the identity commutes with any operator and $P_2 = \mathbf {1}- P_1$ (see (D.4)), we prove the lemma for $P_1$ only. Since $h_{\mathrm{per}}$ (see (3.8)) has a gap at $\mu$, the Cauchy integral formula implies

$$\begin{aligned} P_1 = \frac{1}{2\pi i} \int _{\Gamma _1} (z-h_{\mathrm{per}})^{-1} = \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) \end{aligned}$$

(D.15)

where $\Gamma _1$ is the contour $\{t + i ; -c\le t< \mu \} \cup \{t - i ; -c\le t < \mu \} \cup \{ -c -it + (1-t)i : t \in [0,1] \} \cup \{ \mu -it + (1-t)i : t \in [0,1] \}$, where $c > 0$ is any constant such that $h_{\mathrm{per}} > -c +1$, and the contour is traversed counter-clockwise. We see that

$$\begin{aligned}{}[P_1, \psi ]&= \frac{1}{2\pi i} \int _{\Gamma _1} [ r_{\mathrm{per}}(z) ,\psi ] \end{aligned}$$

(D.16)

$$\begin{aligned}&= \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) [\nabla \cdot , \nabla \psi ] r_{\mathrm{per}}(z) \nonumber \\&\quad + \frac{1}{2\pi i} \int _{\Gamma _1} r_{\mathrm{per}}(z) (2\nabla \psi \cdot \nabla ) r_{\mathrm{per}}(z). \, \end{aligned}$$

(D.17)

Lemma D.4 is now proved by an application of the Kato–Seiler–Simon inequality ((2.14)) to (D.17) and noting that $\Gamma _1$ is compact and has length O(1). $\square$

Using Lemma D.4 and estimates (3.6) and (D.8) in (D.12) yields that

$$\begin{aligned} \Vert {\text {den}}[N^{(121)}_2(\psi )] \Vert _{L^2} \lesssim \Vert \nabla \psi \Vert _{L^2}^2 \, . \end{aligned}$$

(D.18)

Case 2: (112), (211), (122), (221) We estimate the case for (112), the other cases are done similarly. Again, since $P_1P_2 = 0$, we write

$$\begin{aligned} N^{(112)}_2(\psi )&= \oint R_1 \psi R_1 \psi R_2 \end{aligned}$$

(D.19)

$$\begin{aligned}&= \oint R_1 \psi R_1 [\psi , P_1] R_2 \, . \end{aligned}$$

(D.20)

Using Lemma 2.1 as in with $N^{(121)}_2(\psi )$ in (D.11), we estimate (D.20) as

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2}&\lesssim \left| \oint \right| d^{-1} \Vert \psi R_1 \Vert \Vert [\psi , P_1]R_2 \Vert _{S^2} . \end{aligned}$$

(D.21)

where $\left| \oint \right|$ is defined in (D.13). By the inequality $\Vert A\Vert \le \Vert A\Vert _{I^p}$, for any $p < \infty$ for any operator A on $L^2(\mathbb {R}^3)$, and the Kato–Seiler–Simon inequality (2.14), we find $\Vert \psi R_1 \Vert \le \Vert \psi R_1 \Vert _{S^6}\lesssim \Vert \psi \Vert _{L^6}$. Using this, together with Lemma D.4, in (D.21), we obtain

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2} \lesssim&\left| \oint \right| d^{-2} \Vert \psi \Vert _{L^6} \Vert \nabla \psi \Vert _{L^2} . \end{aligned}$$

(D.22)

Combining this with (3.6), (D.8) and Hardy–Littlewood’s inequality (2.18) gives

$$\begin{aligned} \Vert {\text {den}}[N^{(112)}_2(\psi )] \Vert _{L^2} \lesssim&\Vert \nabla \psi \Vert _{L^2}^2. \end{aligned}$$

(D.23)

Case 3 (111) and (222). We use the $L^2$–$L^2$ duality to estimate the $L^2$ norm of ${\text {den}}[N^{(qqq)}_2(\psi )], q=1, 2$. We have, by (2.5) and definition (5.12),

$$\begin{aligned} \Vert {\text {den}}[N^{(qqq)}_2(\psi )]\Vert _{L^2}&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \int f {\text {den}}[N^{(qqq)}_2(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \mathrm {Tr}[f N^{(qqq)}_2(\psi )]\right| \nonumber \\&=\sup _{ \Vert f\Vert _{L^2}=1}\left| \oint \mathrm {Tr}(f R_q \psi R_q \psi R_q)\right| . \end{aligned}$$

(D.24)

(In the last two lines, f is considered as a multiplication operator.) To show that the integral on the r.h.s. converges absolutely, we follow the arguments in (5.17)–(5.20) to prove, for $q=0, 1$,

$$\begin{aligned} \big |\mathrm {Tr}( f R_q (\psi R_q)^2)\big |\lesssim&d^{-4/3}\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.25)

Due to definition (D.8) of $d \equiv d(z)$, this shows that the integral on the r.h.s. of (D.24) converges absolutely.

Lemma D.5

For $q=1, 2$, we have

$$\begin{aligned} \oint&\mathrm {Tr}[f R P_q g R P_q h P_q] \nonumber \\&= \frac{1}{2\pi i}\int _{\Gamma _q} (f_{T}(z) - 1) \mathrm {Tr}[f R P_q g R P_q h P_q]. \end{aligned}$$

(D.26)

Proof

Note that the contour $\Gamma$ in Fig. 2 is the union of two disjoint contours, $\Gamma =\Gamma _1 \cup \Gamma _2$, with $\Gamma _1$ being the closed contour and $\Gamma _1$ unbounded one (i.e. the parts of $\Gamma$ with $\mathrm {Re}\, z < \mu$ and $\mathrm {Re}\, z > \mu$). We first note that, by Bloch’s theory,

$$\begin{aligned} \int _{\Gamma _q} dz \,&\mathrm {Tr}[f R P_q g R P_q h P_q] =\int _{\Gamma _q} dz\, \int _{(\Omega ^*)^3} d{\hat{k}} d{\hat{k}}_1 d {\hat{k}}_2 \end{aligned}$$

(D.27)

$$\begin{aligned}&\times \mathrm {Tr}_{L^2_{\mathrm{per}}} f_{k-k_1} (RP_q)_{k_1} g_{k_1-k_2} (RP_q)_{k_2} h_{k_2-k} (RP_q)_k . \end{aligned}$$

(D.28)

Computing the trace in the complete orthonormal basis of eigenvectors $\varphi _{m,k}$ of $(RP_q)_{k}$ (with eigevalues $\lambda _{m,k}$) and inserting the complete orthonormal bases of eigenvectors $\varphi _{n,k_1}$ and $\varphi _{r, k_2}$ of $(RP_q)_{k_1}$ and $(RP_q)_{k_2}$ (with eigevalues $\lambda _{n,k_1}$ and $\lambda _{r,k_2}$) into (D.28), we see that

$$\begin{aligned} \int _{\Gamma _1}&\mathrm {Tr}[f R P_q g R P_q h P_q] = \sum _{m,n,r} \int _{(\Omega ^*)^3} d{\hat{k}} d{\hat{k}}_1 d {\hat{k}}_2 \end{aligned}$$

(D.29)

$$\begin{aligned}&\times \langle \varphi _{m,k}, f_{k-k_1} \varphi _{n, k_1} \rangle \langle \varphi _{n, k_1}, g_{k_1-k_2} \varphi _{r, k_2} \rangle \langle \varphi _{r, k_2}, h_{k_2-k} \varphi _{m, k} \rangle \end{aligned}$$

(D.30)

$$\begin{aligned}&\times \int _{\Gamma _q} dz\, \frac{1}{(z-\lambda _{m,k})(z-\lambda _{n,k_1})(z-\lambda _{r, k_2})} . \end{aligned}$$

(D.31)

Since $P_1$ projects to the spectrum of $h_{\mathrm{per}}$ on the left of $\mu$, we see that $\lambda _{m,k}, \lambda _{n,k_1}, \lambda _{r, k_2} < \mu$. In particular, these eigenvalues are in the left closed contour in Fig. 2. Consequently, Cauchy’s integral formula shows that the term in the large bracket in (D.31) is identically zero. Similar argument applies to $P_2$. This shows that

$$\begin{aligned} \frac{1}{2\pi i}\int _{\Gamma _q} \mathrm {Tr}[f R P_q g R P_q h P_q] = 0. \end{aligned}$$

(D.32)

Thus (D.26) follows. $\square$

Using the explicit form of the Fermi–Diract distribution $f_{T}$ in (1.2), we see that

$$\begin{aligned} |f_{T}(z-\mu ) - 1| = \frac{e^{\beta (\mathrm {Re}\, z-\mu )}}{|1+e^{\beta (z-\mu )}|} . \end{aligned}$$

(D.33)

By condition (1.26) and by the choice of the contour, $\Gamma$, in Fig. 2, we see that the if $z \in \Gamma$, then $\mathrm {Re}\, z$ is at least at the distance $\ge 1$ from $\mu$. Hence, for z in a contour $\Gamma$, (D.33) implies that

$$\begin{aligned} |f_{T}(z-\mu ) - 1| \lesssim e^{-\beta }. \end{aligned}$$

(D.34)

Applying estimates (D.34) and (D.25) to the r.h.s. of (D.26) and recalling the definition (D.8) of $d\equiv d(z)\ge \frac{1}{4}$,

we arrive at the inequality

$$\begin{aligned}&\left| \oint \mathrm {Tr}[f R P_q g R P_q h P_q]\right| \nonumber \\&\quad \lesssim e^{-\beta }\Vert f\Vert _{L^2} \Vert \nabla \psi \Vert _{L^2}^{4/3} \Vert \psi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.35)

This inequality, together with the relation (D.24), gives

$$\begin{aligned} \left\| {\text {den}}[N^{(qqq)}_2(\phi )]\right\| _{L^2} \lesssim&e^{-\beta } \Vert \nabla \phi \Vert _{L^2}^{4/3} \Vert \phi \Vert _{L^2}^{2/3}. \end{aligned}$$

(D.36)

Inequalities (D.18), (D.23) and (D.36) imply estimate (D.3) for $k=2$.

Now we estimate $N_k$ for $k>2$. By (3.6) and (3.10), it suffices to estimate ${\text {den}}[R(\phi R)^k]$ where $R = r_{\mathrm{per}}(z)$ is given in (3.8). Using Lemma 2.1, we see that

$$\begin{aligned} \Vert {\text {den}}[R(\phi R)^k]\Vert _{L^2} \lesssim&\Vert (1-\Delta )^{4/3+\epsilon }R(\phi R)^k\Vert _{S^2} \end{aligned}$$

(D.37)

$$\begin{aligned} \lesssim&\Vert (\phi R)^k\Vert _{S^2}. \end{aligned}$$

(D.38)

Using Hölder’s inequality with $\frac{1}{2} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} +$ another k terms of $\frac{1}{\infty }$, (D.38) becomes

$$\begin{aligned} \Vert {\text {den}}[R(\phi R)^k]\Vert _{L^2} \le&\Vert \phi R\Vert _{S^6}^3 \Vert \phi R\Vert ^{k-3} \end{aligned}$$

(D.39)

$$\begin{aligned} \le&\Vert \phi R\Vert _{S^6}^k , \end{aligned}$$

(D.40)

where the last line follows since $\Vert \cdot \Vert \le \Vert \cdot \Vert _{S^p}$ for $p < \infty$. Combining with Kato–Seiler–Simon’s inequality (2.14) and Hardy-Littlewood’s inequality (2.18), (D.40) implies (D.3) for $k\ge 3$. $\square$

The rest of the proof of Proposition D.1 proceeds as the proof of in Proposition 5.3. $\square$