Microlocal Analysis and Adiabatic Problems: The Case of Perturbed Periodic Schrödinger Operators

Abstract

Microlocal analysis is a powerful technique to deal with multiscale and adiabatic problems in Quantum Mechanics. We illustrate this general claim in the specific case of a perturbed periodic Schrödinger operator, namely the operator defined in a dense subspace of ${L}^{2}({\mathbb{R}}^{d})$ by

$$ {H}_{\epsilon } = \frac{1} {2}{\sum \limits }_{j=1}^{d}{\left (-i \frac{\partial } {\partial {x}_{j}} - {A}_{j}(\epsilon x)\right )}^{2} + V (x) + \varphi (\epsilon x), $$

(1)

where $V : {\mathbb{R}}^{d} \rightarrow \mathbb{R}$ is a ${\mathbb{Z}}^{d}$-periodic function, $V \in {L}_{\mathrm{loc}}^{2}({\mathbb{R}}^{d})$, corresponding to the interaction of the test electron with the ionic cores of a crystal, while ${A}_{j} \in {C}_{\mathrm{b}}^{\infty }({\mathbb{R}}^{d})$ and $\varphi \in {C}_{\mathrm{b}}^{\infty }({\mathbb{R}}^{d})$ represent some perturbing external electromagnetic potentials.

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Microlocal analysis is a powerful technique to deal with multiscale and adiabatic problems in Quantum Mechanics. We illustrate this general claim in the specific case of a perturbed periodic Schrödinger operator, namely the operator defined in a dense subspace of ${L}^{2}({\mathbb{R}}^{d})$ by

$$ {H}_{\epsilon } = \frac{1} {2}{\sum \limits }_{j=1}^{d}{\left (-i \frac{\partial } {\partial {x}_{j}} - {A}_{j}(\epsilon x)\right )}^{2} + V (x) + \varphi (\epsilon x), $$

(1)

where $V : {\mathbb{R}}^{d} \rightarrow \mathbb{R}$ is a ${\mathbb{Z}}^{d}$-periodic function, $V \in {L}_{\mathrm{loc}}^{2}({\mathbb{R}}^{d})$, corresponding to the interaction of the test electron with the ionic cores of a crystal, while ${A}_{j} \in {C}_{\mathrm{b}}^{\infty }({\mathbb{R}}^{d})$ and $\varphi \in {C}_{\mathrm{b}}^{\infty }({\mathbb{R}}^{d})$ represent some perturbing external electromagnetic potentials. The parameter $\epsilon \ll 1$ corresponds to the separation of space-scales.

Since the unperturbed Hamiltonian ${H}_{\mathrm{per}} = -\frac{1} {2}\Delta + V$ is periodic, it can be decomposed as a direct integral of simpler operators, thus exhibiting a band structure, analogous to the one appearing in the Born-Oppenheimer problem.

We are interested to the behavior of the solutions to the dynamical Schrödinger equation $i\epsilon \,{\partial }_{t}{\psi }_{\epsilon }(t) = {H}_{\epsilon }{\psi }_{\epsilon }(t)$ in the limit $\epsilon \rightarrow 0$. We show that by using microlocal analysis with operator-valued symbols one can decouple the dynamics corresponding to different bands and determine a simpler approximate dynamics for each band [3]. Further developments have been obtained, more recently, in [1, 5].

The Bloch-Floquet transform. The ${\mathbb{Z}}^{d}$-symmetry of the unperturbed Hamiltonian operator ${H}_{\mathrm{per}} = -\frac{1} {2}\Delta + V$ can be used to decomposed it as a direct integral of simpler operators. To fix the notation, let Y be a fundamental domain for the action of the translation group $\Gamma = {\mathbb{Z}}^{d}$ on ${\mathbb{R}}^{d}$, and let $\mathbb{B}$ be a fundamental domain for the action of the dual lattice ${\Gamma }^{{_\ast}} := \left \{\kappa \in {({\mathbb{R}}^{d})}^{{_\ast}} : \kappa \cdot \gamma \in 2\pi \mathbb{Z}\quad \forall \gamma \in \Gamma \right \}$ on the dual space ${({\mathbb{R}}^{d})}^{{_\ast}}$ (“momentum space”). We also introduce the tori ${\mathbb{T}}_{Y }^{d} = {\mathbb{R}}^{d}/\Gamma$ and ${\mathbb{T}}^{{_\ast}} ={ (\mathbb{R}}^{d}{)}^{{_\ast}}/{\Gamma }^{{_\ast}}$. The formula

$$(\widetilde{\mathcal{U}}\psi )(k,y) ={ \sum \limits }_{\gamma \in \Gamma }\mathrm{{e}}^{-ik\cdot (y+\gamma )}\,\psi (y + \gamma ),\qquad y \in {\mathbb{R}}^{d},k \in {({\mathbb{R}}^{d})}^{{_\ast}},\psi \in \mathcal{S}({\mathbb{R}}^{d})$$

extends to a unitary operator $\widetilde{\mathcal{U}} : {L}^{2}({\mathbb{R}}^{d})\rightarrow {L}^{2}(\mathbb{B}) \otimes {L}^{2}({\mathbb{T}}_{Y }^{d}) \simeq {L}^{2}(\mathbb{B},{L}^{2}({\mathbb{T}}_{Y }^{d}))$, called (modified) Bloch-Floquet transform. Hereafter ${\mathcal{H}}_{\mathrm{f}} := {L}^{2}({\mathbb{T}}_{Y }^{d})$.

The advantage of this construction is that, after conjugation, H _per becomes a fibered operator, namely

$$\begin{array}{rcl} \widetilde{{H}}_{\mathrm{per}} :=\widetilde{ \mathcal{U}}\,{H}_{\mathrm{per}}\,\widetilde{{\mathcal{U}}}^{-1} ={ \int \limits }_{\mathbb{B}}^{\oplus }{H}_{\mathrm{ per}}(k)\,dk& & \mbox{ in }{L}^{2}(\mathbb{B},{\mathcal{H}}_{\mathrm{ f}}) \simeq {\int \limits }_{\mathbb{B}}^{\oplus }{\mathcal{H}}_{\mathrm{ f}}\,dk =: \mathcal{H}, \\ {H}_{\mathrm{per}}(k) = \dfrac{1} {2}{(-i{\nabla }_{y} + k)}^{2} + V (y)& & \text{ acting on}\,\,\,\mathcal{D}\subseteq {L}^{2}({\mathbb{T}}_{ Y }^{d},dy) = {\mathcal{H}}_{\mathrm{ f}}\, \\ \end{array}$$

where $\mathcal{D}$ is a dense subspace of ${\mathcal{H}}_{\mathrm{f}}$. The operator H _per(k) has compact resolvent, and we label its eigenvalues as ${E}_{0}(k) \leq {E}_{1}(k) \leq \ldots$. Notice that the eigenvalues are ${\Gamma }^{{_\ast}}$-periodic. We assume that a solution of the eigenvalue problem ${H}_{\mathrm{per}}(k)\,{\chi }_{n}(k,y) = {E}_{n}(k)\,{\chi }_{n}(k,y)$ is known, and we denote by P _n (k) the eigenprojector corresponding to the n-th eigenvalue, while ${P}_{n} ={ \int \limits }_{\mathbb{B}}^{\oplus }{P}_{n}(k)\,dk$. The set ${\mathcal{E}}_{n} =\{ (k,{E}_{n}(k)) \in {\mathbb{T}}^{{_\ast}}\times \mathbb{R}\}$ is called the n-th Bloch band.

The perturbed dynamics. We consider a Bloch band ${\mathcal{E}}_{n}$ which is separated by a gap from the rest of the spectrum, i. e.

$$\inf \{\vert {E}_{n}(k) - {E}_{m}(k)\vert : k \in {\mathbb{T}}^{{_\ast}},m\neq n\} > 0,$$

(2)

and the corresponding subspace

$$\mathrm{Ran}{P}_{n} =\{ \Psi \in \mathcal{H} : \Psi (k,y) = \varphi (k)\,{\chi }_{n}(k,y)\mbox{ for }\varphi \in {L}^{2}(\mathbb{B},dk)\}.$$

In the unperturbed case, A = 0 and ϕ = 0, the subspace RanP _n is exactly invariant, in the sense that $(1 - {P}_{n})\,{e}^{-i\widetilde{{H}}_{\mathrm{per}}t/\epsilon }\,{P}_{n}\Psi = 0$ for all $\Psi \in \mathcal{H}.$ Moreover, the dynamics of $\Psi \in \mathrm{ Ran}{P}_{n}$ is particularly simple, namely

$$\left ({e}^{-i\widetilde{{H}}_{\mathrm{per}}t/\epsilon }\Psi \right )(k,y) = \left ({e}^{-i{E}_{n}(k)t/\epsilon }\varphi (k)\right ){\chi }_{ n}(k,y).$$

Thus a natural question arises: to what extent such properties survive in the perturbed case? More precisely,

• (i) Does exist a subspace of $\mathcal{H}$ which is almost-invariant with respect to the dynamics, up to errors of order ${\epsilon }^{N}$?

• (ii) Is there any simple (and numerically convenient) way to approximately describe the dynamics inside the almost invariant subspace?

The microlocal approach. Microlocal analysis is a useful tool to answer these questions. In a nutshell, one checks that by modified BF transform one has

$$\widetilde{{H}}_{\epsilon } :=\widetilde{ \mathcal{U}}\,\,{H}_{\epsilon }\,\,\widetilde{{\mathcal{U}}}^{-1} ={ \left (-i{\nabla }_{ y} + k - A(i\epsilon {\nabla }_{k})\right )}^{2} + V (y) + \phi (i\epsilon {\nabla }_{ y}).$$

The latter operator “looks like” the $\epsilon$-Weyl quantization of an operator-valued symbol

$$\begin{array}{rcl} h :& {\mathbb{T}}^{{_\ast}}\times {\mathbb{R}}^{d}& \rightarrow \mathrm{Operators}(\,{\mathcal{H}}_{\mathrm{f}}\,) \\ & (k,r) & \longmapsto {\left (-i{\nabla }_{y} + k - A(r)\right )}^{2} + V (y) + \phi (r).\end{array}$$

This observation naturally leads to exploit techniques related to matrix-valued pseudo-differential operators [2, 4]. Obviously, to perform this program one has to circumvent some technical scholia (unbounded-operator-valued symbols, covariance, …), for whose solution we refer to [3]. As an answer to question (i), we have the following

FormalPara Theorem 1.

Let ${\mathcal{E}}_{n}$ be an isolated Bloch band, see (2). Then there exists an orthogonal projection ${\Pi }_{n,\epsilon } \in \mathcal{B}(\mathcal{H})$ such that for every $N \in \mathbb{N}$ there exist C _N such that

$${\left \|[\widetilde{{H}}_{\epsilon },{\Pi }_{n,\epsilon }]\right \|}_{\mathcal{B}(\mathcal{H})} \leq {C}_{N}\,{\epsilon }^{N}$$

and ${\Pi }_{n,\epsilon }$ is $\mathcal{O}({\epsilon }^{\infty })$ -close to the $\epsilon$ -Weyl quantization of a symbol with principal part ${\pi }_{0}(k,r) = {P}_{n}(k - A(r))$.

As for question (ii), one preliminarily notices that there is no natural identification between $\mathrm{Ran}{\Pi }_{n,\epsilon }$ and ${L}^{2}({\mathbb{T}}^{{_\ast}},dk)$, so no evident reduction of the number of degrees of freedom. To circumvent this obstacle, one constructs an intertwining unitary operator (which is an additional unknown in the problem) ${U}_{n,\,\epsilon } :\mathrm{ Ran}{\Pi }_{n,\,\epsilon } \rightarrow {L}^{2}({\mathbb{T}}^{{_\ast}},dk)$. The freedom to choose ${U}_{n,\,\epsilon }$ can be exploited to obtain a simple and physically transparent representation of the dynamics, as in the following result [3].

FormalPara Theorem 2.

Let ${\mathcal{E}}_{n}$ be an isolated Bloch band. Define the effective Hamiltonian as the operator $\hat{{H}}_{\mathrm{eff,\,\epsilon }} := {U}_{n,\,\epsilon }\,{\Pi }_{n,\,\epsilon }\,{H}_{\epsilon }\,{\Pi }_{n,\,\epsilon }\,{U}_{n,\,\epsilon }^{-1}$ acting in ${L}^{2}({\mathbb{T}}^{{_\ast}},dk)$ . Then:

• (i) (approximation of the dynamics) for any $N \in \mathbb{N}$ there is C _N such that

  $${\left \|\left ({\epsilon }^{-i\widetilde{{H}}_{\epsilon }t/\epsilon } - {U}_{ n,\,\epsilon }^{-1}\,\,{\epsilon }^{-i\,\hat{{H}}_{\mathrm{eff,\,\epsilon }}\,\,t/\epsilon }\,{U}_{ n,\,\epsilon }\right ){\Pi }_{n,\,\epsilon }\right \|}_{\mathcal{B}(\mathcal{H})} \leq {C}_{N}\,{\epsilon }^{N}\,(1 + \vert t\vert ).$$

• (ii) (explicit description of the approximated dynamics) the operator $\hat{{H}}_{\mathrm{eff,\epsilon }}$ is $\mathcal{O}({\epsilon }^{\infty })$ -close to the $\epsilon$ -Weyl quantization of the symbol ${h}_{\epsilon }^{\mathrm{eff}} : {\mathbb{T}}^{{_\ast}}\times {\mathbb{R}}^{d} \rightarrow \mathbb{C}$ , with leading orders

  $$\begin{array}{lcl} \ \ \ {h}_{0}^{\mathrm{eff}}(k,r)& =&{E}_{n}(k - A(r)) + \phi (r)\\ & & \\ \ \ \ {h}_{1}^{\mathrm{eff}}(k,r)& =&\left (\nabla \phi (r) -\nabla {E}_{n}(\kappa )\wedge B(r)\right ) + {\mathcal{A}}_{n}(\kappa ) - B(r) \cdot {M}_{n}(\kappa )\end{array}$$

  where κ(k,r) = k − A(r), ${B}_{jl} = {\partial }_{j}{A}_{l} - {\partial }_{l}{A}_{j}$, ${\mathcal{A}}_{n}(k) = i{\left \langle {\chi }_{n}(k)\,\vert \,\nabla {\chi }_{n}(k)\right \rangle }_{{\mathcal{H}}_{\mathrm{f}}}$ is called Berry connection and

  $${M}_{n}(k) = \frac{i} {2}{\left \langle \nabla {\chi }_{n}(k)\wedge \,\vert \,({H}_{\mathrm{per}}(k) - {E}_{n}(k))\nabla {\chi }_{n}(k)\right \rangle }_{{\mathcal{H}}_{\mathrm{f}}}.$$