DISCRETE SPECTRUM ASYMPTOTICS FOR THE TWO-PARTICLE SCHRÖDINGER OPERATOR ON A LATTICE

Abstract

We consider a Hamiltonian of a system of two fermions on a two-dimensional lattice $\mathbb {Z}^2$ with a potential of a certain type. The Schrödinger operator $H(\textbf{k}),\textbf{k}\in \mathbb {T}^2,$ corresponding to the considered system has two invariant subspaces $L_2^{oe}(\mathbb { T }^2)$ and $L_2^{eo}(\mathbb {T}^2)$ in the space of odd functions $L_2^{o}(\mathbb {T}^2),$ where the restrictions of the operator $H(\textbf{k})$ in ${L_2^{oe}(\mathbb {T}^2)}$ and ${L_2^{eo}(\mathbb {T}^2)}$ are denoted by $H^{oe}(\textbf{k})$ and $H^{eo}(\textbf{k})$, respectively. It is proved that for $\textbf{k}_{\beta }=(\pi -2\beta ,\pi ),$ the operator $H^{oe}(\textbf{k}_{\beta }),$ has a finite number of eigenvalues for any $\,\,\beta \in (0,\pi ),$ and for any $\beta \in [0,\pi ]$ operator $H^{eo}(\textbf{k}_{\beta })$ has an infinite number of eigenvalues. It is shown that if $\beta \rightarrow 0,$ then the number of eigenvalues $N(\beta )$ of the operator $H^{oe}(\textbf{k}_{\beta })$ tends to infinity and the following asymptotic formula $\underset{\beta \rightarrow 0}{\lim }\dfrac{N(\beta )}{|\log \sin \beta |}=2$ holds.

Introduction and the main results

Lattice models play an important role in various branches of physics. Among such models are the lattice few-body Hamiltonians [1] that may be viewed as a minimalist version of the corresponding Bose– or Fermi–Hubbard model involving a fixed finite number of particles of a certain type. Surely, the few-body lattice energy operators are of a great theoretic interest already in their own right [2,3,4,5]. In addition, these discrete Hamiltonians may be viewed as a natural approximation for their continuous counterparts [6] allowing to study few-body phenomena in the context of the theory of bounded operators.

The discrete Schrödinger operators represent the simplest and natural model for description of fewbody systems formed by particles traveling through periodic structures, say, for ultracold atoms injected into optical crystals created by the interference of counter-propagating laser beams [7, 8]. The study of ultracold few-atom systems on optical lattices became very popular in the last years since these systems possess highly controllable parameters such as lattice geometry and dimensionality, particle masses, two-body potentials, temperature, etc (see, e.g., [7, 9, 10] and references therein). Unlike the traditional condensed matter systems, where stable composite objects are usually formed by attractive forces, the controllability of the ultracold atomic systems on an optical lattice gives an opportunity to experimentally observe a stable repulsive bound pair of ultracold atoms, see, e.g., [8, 11]. The main difficulty in solving Fermi–Hubbard Hamiltonian even with the on-site interaction is due to the tunneling, i.e., the kinetic energy necessary for a fermion to hop from site to site, since it is highly nonlocal. Moreover, unlike its continuous counterpart, the lattice Hamiltonian, corresponding to short-range interacting systems of particle pairs, is not rotationally invariant. Hence, the separation of the lattice energy operator related to the center of mass motion is not possible.

Unlike in the continuous case, the lattice few-body system does not admit separation of the center of mass motion. However, the discrete translation invariance allows one to use the Floquet–Bloch decomposition. In particular, the total two-particle lattice energy operator H in the momentum representation may be written as the von Neumann direct integral

$$H\simeq \int _{{\textbf{k} \in \mathbb {T}}^2}\oplus H(\textbf{k}) \,\textrm{d}\textbf{k},$$

where $\mathbb {T}^2$ is the two-dimensional torus and $H(\textbf{k})$ is the fiber operator acting in the respective functional Hilbert space on $\mathbb {T}^2$. Recall that the fiber operators $H(\textbf{k})$ nontrivially depend on the quasimomentum $\textbf{k}$ (see, e.g., [1, 12,13,14]) and spectra of the family $H(\textbf{k})$ become very sensitive to changes in $\textbf{k}\in \mathbb {T}^2$ (see Theorem 1.2).

The operator $H(\textbf{k }), \,\, \textbf{k } \in {\mathbb {T} ^2}$, corresponding to a system of two fermions on a two-dimensional lattice, acting in the Hilbert space of odd functions $L_2^{o}({{{\mathbb {T}}^2}})$ by the form:

$$\begin{aligned} H(\textbf{k}) =H_{0}(\textbf{ k })-V, \end{aligned}$$

(1.1)

where

$$\begin{aligned}({H}_{0}(\textbf{k})f)(\textbf{q})= \varepsilon _\textbf{k} (\textbf{q})f(\textbf{q}), \quad (Vf)(\textbf{q})=\frac{1}{2\pi }\int _{{\mathbb {T}}^2} v (\mathbf {q-s})f(\textbf{s})\,\textrm{d}\textbf{s}, \end{aligned}$$

$$\begin{aligned} \varepsilon _{\textbf{k}}(\textbf{q})= \varepsilon (\frac{\textbf{k}}{2}+\textbf{q}) +\varepsilon (\frac{\textbf{k}}{2}-\textbf{q}); \quad \varepsilon (\textbf{q})=\sum _{i=1}^{2}\left( 1-\cos q_i\right) , \quad \textbf{q}=(q_1, q_2) \in \mathbb {T}^2. \end{aligned}$$

(1.2)

The kernel of the integral operator V has the form

$$\begin{aligned} v (\mathbf {q-s})=\frac{1}{2\pi }\sum _{\textbf{n}\in \mathbb {{Z}}^2}\hat{v}(\textbf{n})e^{i(\mathbf {q-s},\textbf{n})}, \end{aligned}$$

where the potential $\hat{v}$ is defined as

$$\begin{aligned} \hat{v}(\textbf{n})=\hat{v}(n_1,n_2)={\left\{ \begin{array}{ll} \bar{v}(\mathbf {|n|})=c_0e^\mathbf {-|n|}, \quad \text{ if } \quad |n_1|\le 2 \\ 0, \qquad \text{ if } \quad |n_1|\ge 3, \end{array}\right. } \end{aligned}$$

(1.3)

$c_0>0$ is some constant, $\mathbf {|n|}=|n_1|+|n_2|.$

Remark 1.1

From the definition of $\hat{v}(\textbf{n})$ it follows that the kernel $v(\textbf{q})$ of the integral operator V is represented as $v(\textbf{q})=\dfrac{c_0}{2\pi }v_1(q_1)v_2(q_2),$ where

$$v_1(q_1)=\sum _{n_1=-2}^{2}e^{in_1q_1-|n_1|}=1+\frac{2}{e}\cos q_1+\frac{2}{e^2}\cos 2q_1,$$

$$v_2(q_2)=\sum _{n_2 \in \mathbb {Z}}^{2}e^{in_2q_2-|n_2|}=1+2\sum _{m=1}^{\infty }e^{-m}\cos mq_2.$$

It is known that the space $L_2^o({\mathbb {T}}^2)$ can be decomposed into the direct sum $L_2^o({\mathbb {T}}^2)= L_2^{oe}({\mathbb {T}}^2)\oplus L_2^{eo}({\mathbb {T}}^2)$ and subspaces $L_2^{oe}({\mathbb {T}}^2)$, $L_2^{eo}({\mathbb {T}}^2)$ are represented as the tensor product

$$L_2^{oe}({\mathbb {T}}^2)=L_2^{o}({\mathbb {T}})\otimes L_2^{e}({\mathbb {T}}),\quad \quad L_2^{eo}({\mathbb {T}}^2)=L_2^{e}({\mathbb {T}})\otimes L_2^{o}({\mathbb {T}}),$$

where

$$L_2^{o}({\mathbb {T}})=\{f\in L_2({\mathbb {T}}): f(-q)=-f({q}) \},\quad \quad L_2^{e}({\mathbb {T}})=\{f\in L_2({\mathbb {T}}): f(-q)=f({q})\}.$$

We remark that the subspaces $L_2^{oe}({\mathbb {T}}^2)$ and $L_2^{eo}({\mathbb {T}}^2)$ are invariant under the operator $H(\textbf{k})$ (see Lemma 3.1). We denote by $H^{oe}(\textbf{k})$, $H^{eo}(\textbf{k})$ the restrictions of the operator $H(\textbf{k})$ in the subspaces $L_2^{oe}({\mathbb {T}}^2)$, $L_2^{eo}({\mathbb {T}}^2)$, respectively, and by $N(\beta )$ the number of eigenvalues of the operator $H^{oe}(\textbf{k}_{\beta }), \,\, \textbf{k}_{\beta }:=(\pi -2\beta ,\pi ), \,\,\beta \in [0, \pi ],$ lying below the essential spectrum.

The following theorem describes an asymptotic formula for $N(\beta )$ as $\beta \rightarrow 0$.

Theorem 1.2

For any $\beta \in (0,\pi )$, the operator $H^{oe}(\textbf{k}_{\beta })$ has a finite number of eigenvalues lying below the essential spectrum such that $N(\beta )$ increases as $\beta \rightarrow 0$ and the equality

$$\begin{aligned} \lim _{\beta \rightarrow 0}\dfrac{N(\beta )}{|\log \sin \beta |}=2 \end{aligned}$$

(1.4)

holds.

Theorem 1.3

For any $\beta \in [0,\pi ]$ the operator $H^{eo}(\textbf{k}_{\beta })$ has an infinite number of eigenvalues lying below the essential spectrum.

Some comments on the main results are in order.

• All discontinuous points $\beta _n^\pm$ of the function $N(\beta )$ are found (see Eq. 4.15) for which the jumps at these points are equal to one.

• If $\textbf{k}=\textbf{k}_{\beta _n^\pm },$ then the lower edge of the essential spectrum is the threshold resonance (see Eq. 4.4) for $H^{oe}(\textbf{k}_{\beta }).$

• The results of the Theorem 1.2 are characteristic for the lattice systems, in fact, they do not have any analogues in the continuous case.

The discrete spectrum of the two-particle continuous Schrödinger operator $h_\lambda =-\Delta +\lambda V$ has been studied by many authors, with the conditions for the potential V formulated in its position. The condition for the finiteness of the set of negative elements of the spectrum and the absence of positive eigenvalues of $h_\lambda$ can be found in [15]. If $V\le 0,$ then the number of negative eigenvalues $N(\lambda )$ is a nondecreasing function of $\lambda \in (0,\infty )$, and each eigenvalue $z_n(\lambda )$ decreases on the half-axis $(0,\infty ).$ It is known that when the coupling constant $\lambda$ decreases, the bound-state energies of $h_\lambda$ tend to the boundary of the continuous spectrum [15] and for some finite $\lambda$ are on the boundary. Two questions then arise: Does a bound or virtual state correspond to such a threshold state (i.e., is the corresponding wave function square-integrable)? And where do the bound states “disappear to”  as $\lambda$ decreases further? The study of the first question was the subject in [16,17,18].

Perturbations of the eigenvalues of a two-particle Shrödinger operator on a one-dimensional lattice were studied in [19]. It is proved that the operator $H(\pi )$ has an infinite number of eigenvalues $z_m( \pi )=\hat{v}(m), m \in \mathbb {Z}_+\,\,$ ($\mathbb {Z}_+=\{0\}\bigcup \mathbb {N}$). If the potential $\hat{v}$ increases in $\mathbb {Z}_+$, then only the eigenvalue $z_0( \pi )$ is simple, and all other eigenvalues have a multiplicity of two. It is proved that for each of the doubly degenerate eigenvalues $z_m( \pi ), \,\,\, m \in \mathbb {N}$ the operator $H( \pi )$ splits into two non-degenerate eigenvalues $z_m^-( k)$ and $z_m^+( k)$ for small changes $k \in (\pi -\delta , \pi ).$ Hamiltonian of a two-particle system on a three-dimensional lattice $\mathbb {Z}^3$. It is proved that the number $N(\textbf{k})=N(k_1,k_2,k_3)$ of eigenvalues H(k) below the essential spectrum of the operator $H(\textbf{k})$ is a non-decreasing function for each $k_i\in [0,\pi ], i = 1, 2, 3.$ In [21] were studied two-particle Schrödinger operator H(k) on the $\nu$–dimensional lattice $\mathbb {Z}^{\nu }$ and proved that the number of negative eigenvalues of H(k) is finite for a wide class of potentials $\hat{v}$.

The work is organized as follows. The “Description of the two-particle Hamiltonian'' section is devoted to the description of a two-particle Hamiltonian on the two-dimensional lattice. In the “Invariant subspaces'' section, invariant subspaces under the operator $H(k_1, \pi )$ are constructed. In addition, it is proved that there exists an infinite number of invariant subspaces with respect to the operators $H^{oe}(\textbf{k}_{\beta })$ and $H^{eo}(\textbf{k}_{\beta })$. In the “On the spectrum of the operator $H^{oe}(k_1,\pi )$'' section, the finiteness of the number of eigenvalues for the operator $H^{oe}(\textbf{k}_{\beta })$ is proved as $\beta \not =0$. The asymptotic formula for the $n-$th eigenvalue $z_n(\beta )$ of $H^{oe}(\textbf{k}_{\beta })$ is found as $\beta \rightarrow 0.$ In the “On the spectrum of the operator $H^{eo}(k_1,\pi )$'' section, it is proved that the operator $H^{eo }( \textbf{k}_{\beta } )$ has an infinite number of eigenvalues for any $\beta$. Moreover, the asymptotic formula for the $n-$th eigenvalue $z_n(\beta )$ of $H^{eo}(\textbf{k}_{\beta })$ is found as $\beta \rightarrow 0.$

Description of the two-particle Hamiltonian

Let $\mathbb {Z}^2$ be a two-dimensional lattice and $\mathbb {Z}^2\times \mathbb {Z}^2$ be the Cartesian product of lattices. Denote by $\ell _2(\mathbb {Z}^2\times \mathbb {Z}^2)$ the Hilbert space of square-summable functions defined on $\mathbb {Z}^2\times \mathbb {Z}^2$ and by $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2)\subset \ell _2(\mathbb {Z}^2\times \mathbb {Z}^2)$ subspace of antisymmetric functions, i.e., $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2):=\{f\in \ell _2(\mathbb {Z}^2\times \mathbb {Z}^2): f(\textbf{x}, \textbf{y})= -f(\textbf{y}, \textbf{x})\}.$ The unperturbed operator $\hat{H}_{0}$ of a system of two fermions with unit mass on two-dimensional $\mathbb {Z}^2$ is a bounded self-adjoint operator acting in the Hilbert space $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2)$ as

$$\hat{H}_0=-\frac{1}{2}\Delta _{1}-\frac{1}{2}\Delta _{2}.$$

Here $\Delta _{1}={\Delta }\otimes I$ and $\Delta _{2}=I\otimes \Delta ,$ where I is the identity operator, the lattice Laplacian $\Delta$ is a difference operator describing the transfer of a particle from one site to a nearest-neighbor site:

$$(\Delta \hat{\psi })(\textbf{x})= \sum _{j=1}^2 [\hat{\psi }(\mathbf {x+e}_j)+\hat{\psi }(\mathbf {x-e}_j)- 2\hat{\psi }(\textbf{x})],\quad \textbf{x}\in \mathbb {Z}^2,\quad \hat{\psi }\in \ell _2(\mathbb {Z}^2),$$

where $\textbf{e}_1=(1,0),\, \textbf{e}_2=(0,1)$ are unit vectors in $\mathbb {Z}^2.$ The total Hamiltonian $\hat{H}$ acts in the Hilbert space $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2)$ and consists of the difference between the unperturbed operator $\hat{H}_0$ and the interaction potential $\hat{V}$ of two particles (see [19]):

$$\begin{aligned} \hat{H} =\hat{H}_{0}- \hat{V}, \end{aligned}$$

$$(\hat{V}\hat{\psi })(\textbf{x,y}) ={\hat{v}(\mathbf {x -y} )\hat{\psi }(\textbf{x,y})},\quad \hat{\psi } \in \ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2),$$

where $\hat{v}(\textbf{x})$ function is defined by the formula (1.3).

Under condition (1.3) the energy operator $\hat{H}$ is bounded and self-adjoined in the space $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2).$

Let $\mathbb {T}^{2}$ be a two-dimensional torus and ${L}_2(\mathbb {T}^2\times \mathbb {T}^2)$ be the Hilbert space of square-integrable functions defined on $\mathbb {T}^2\times \mathbb {T}^2,$ $L_2^{as}(\mathbb {T}^2\times \mathbb {T}^2) \subset L_2(\mathbb {T}^2\times \mathbb {T}^2)$ be the subspace of antisymmetric functions. Let $F:\ell _2(\mathbb {Z}^2\times \mathbb {Z}^2)\rightarrow L_2(\mathbb {T}^2\times \mathbb {T}^2)$ be the standard Fourier transform. Let us denote by $\hat{F}:\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2)\rightarrow L_2^{as}(\mathbb {T}^2\times \mathbb {T}^2)$ the restriction F in $\ell _2^{as}(\mathbb {Z}^2\times \mathbb {Z}^2).$ The Hamiltonian $H=H_0-V= \hat{F}\hat{H}\hat{F}^{-1}$ in the momentum representation commutes with the unitary operators $U_\textbf{s},\quad \textbf{s}\in \mathbb {Z}^2:$

$$(U_\textbf{s}f)(\textbf{k}_1,\textbf{k}_2)= \exp \big (-i(\textbf{s},\textbf{k}_1+\textbf{k}_2)\big )f(\textbf{k}_1,\textbf{k}_2),\quad f\in L_2^{as}(\mathbb {T}^2\times \mathbb {T}^2).$$

For any $\textbf{k} \in \mathbb {T}^2$, we set ${\mathcal {F}}_\textbf{k}=\{(\textbf{k}_1,\textbf{k}_2) \in {\mathbb {T}}^2\times {\mathbb {T}}^2: \, \textbf{k}_1+\textbf{k}_2=\textbf{k} \}.$

There exist decompositions of $L_2^{as}({\mathbb {T}}^2\times {\mathbb {T}}^2)$ and the operators $U_\textbf{s},\, H$ into direct integrals (see [12, 15]):

$$L_2^{as}({\mathbb {T}}^2\times {\mathbb {T}}^2)= \int _{{\mathbb {T}}^2} \oplus L_2^{as}({\mathcal {F}}_\textbf{k})\,\textrm{d}\textbf{k}, \quad U_\textbf{s}= \int _{ {\mathbb {T}}^2} \oplus U_\textbf{s}(\textbf{k})\,\textrm{d}\textbf{k}, \quad H= \int _{{\mathbb {T}}^2}\oplus \tilde{H}(\textbf{k}) \,\textrm{d}\textbf{k}.$$

Here $(U_\textbf{s}(\textbf{k})f)(\textbf{q})=e^{-i(\textbf{s,k})}f(\textbf{q})$ with $\textbf{s} \in \mathbb {Z}^2, \textbf{k} \in {{{\mathbb {T}}^2}}.$ Fiber operator $\tilde{H}(\textbf{k})$ of the operator H also acts on $L_2^{as}(\mathcal {F}_\textbf{k})$ and is unitarily equivalent to the operator $H(\textbf{k })$ acting in the Hilbert space of odd functions $L_2^{o}({{{\mathbb {T}}^2}}),$ by the formula (1.1).

The spectra of the operators $H_0(\textbf{k})$ and V are known. The spectrum of $H_0(\textbf{k})$ coincides with the range of the function $\varepsilon _{\textbf{k}}$:

$$\sigma (H_0(\textbf{k}))=[m(\textbf{k}),M(\textbf{k})],$$

where

$$m(\textbf{k})=\min _{\textbf{q}\in {\mathbb {T}}^2}\varepsilon _{\textbf{k}}(\textbf{q})= \varepsilon _{\textbf{k}}(\textbf{0})=2\,\varepsilon \left( {{\frac{\textbf{k}}{2}}}\right) ,\;\quad M(\textbf{k})=\max _{\textbf{q}\in {\mathbb {T}}^2}\varepsilon _{\textbf{k}}(\textbf{q})= \varepsilon _{\textbf{k}}(\varvec{\pi })=8-2\,\varepsilon \left( {{\frac{\textbf{k}}{2}}}\right) .$$

The spectrum of the operator V consists of the set $\{0, \bar{v}(n),\; n\in \mathbb {N}\},$ where $\bar{v}(n)$ is the eigenvalue of the operator V. By assumption (1.3), the operator V is the Hilbert-Schmidt operator. Since V is compact operator by Weyl’s theorem (see [15]) the essential spectrum of the operator $H(\textbf{k})$ coincides with the spectrum of the operator $H_0(\textbf{k}),$ i.e.,

$$\sigma _{ess}(H(\textbf{k}))=[m(\textbf{k}),M(\textbf{k})].$$

By $w(\textbf{k})$, we denote the width of the essential spectrum of $H(\textbf{k}).$ Then,

$$\begin{aligned} w(\textbf{k})=M(\textbf{k})-m(\textbf{k})=8-4\,\varepsilon \left( {\mathbf {\frac{k}{2}}}\right) = 4\cos \frac{k_1}{2}+4\cos \frac{k_2}{2} \end{aligned}$$

(2.1)

and

$$\min _{\textbf{k}\in {\mathbb {T}}^2}w(\textbf{k})=w(\varvec{\pi })=0,\quad \max _{\textbf{k}\in {\mathbb {T}}^2}w(\textbf{k})=w(\varvec{0})=8,$$

where $\varvec{\pi }=(\pi ,\pi ), \quad \textbf{0}=(0,0).$

It can be seen from Eq. 2.1 that if the coordinate $k_j\in [0,\pi ], j=1,2,$ of the total quasimomentum $\textbf{k}$ of the system of two particles increases, then the width of the essential spectrum $w(\textbf{k})$ decreases. We define the width of the essential spectrum of $H(\textbf{k})$ in the direction $\textbf{e}_j, j=1,2,$ by the form

$$\begin{aligned} w_j(\textbf{k})=\max _{p_j\in [-\pi ,\pi ]} \varepsilon _{\textbf{k}}(\textbf{p})-\min _{p_j\in [-\pi ,\pi ]} \varepsilon _{\textbf{k}}(\textbf{p})=4\cos \frac{k_j}{2}, \quad j=1,2. \end{aligned}$$

We have

$$w(\textbf{k})=w_{1}(\textbf{k})+w_{2}(\textbf{k}).$$

If $\textbf{k}=\varvec{\pi },$ then the essential spectrum concentrates at one point $\{4\}$, i.e., $w(\varvec{\pi })=0.$ The spectrum of $H(\varvec{\pi })=4I-V$ consists of eigenvalues of the form $4-\bar{v}(n),\; n\in \mathbb {N}$. If the width of the essential spectrum decreases, then the number of eigenvalues of the Schrödinger operator $H(\textbf{k})$ increases. In particular, if the potential $\hat{v}$ is determined by Eq. 1.3, then the operator $H(k_1,\pi )$ has an infinite number of eigenvalues outside the essential spectrum.

For any self-adjoint operator B acting in a Hilbert space $\mathfrak {{H}}$, we denote by $n[\mu ,B]$ as the number of its eigenvalues lying above $\mu$, $\mu > \sup \sigma _{ess}(B)$.

By the self-adjointness of the operator $H(\textbf{k})=H_{0}(\textbf{ k })-V$ and positivity of $V\;$ it follows that

$$\sigma (H(\textbf{k}))\cap (M(\textbf{k} ),\infty )=\emptyset ,$$

then $\sigma _{disc}(H(\textbf{k}))\subset (-\infty , m(\textbf{k})).$ Therefore, we are only looking for eigenvalues in the interval $(-\infty , m(\textbf{k}))$.

Invariant subspaces

First, we prove the lemma about invariant subspaces with respect to $H(\textbf{k})$.

Lemma 3.1

The subspaces $L_2^{oe}({\mathbb {T}}^2)$ and $L_2^{eo}({\mathbb {T}}^2)$ are invariant under the operator $H(\textbf{k})$.

Proof

Firstly, we prove the invariance of the subspace $L_2^{oe}({\mathbb {T}}^2)$ with respect to $H_0(\textbf{k})$ and V. From the representation (1.2) it follows that $\, \varepsilon _{\textbf{k}} \,$ belongs to the subspace $L_2^{ee}({\mathbb {T}}^2)$, where $L_2^{ee}({\mathbb {T}}^2)=L_2^{e}({\mathbb {T}})\otimes L_2^{e}({\mathbb {T}})$, therefore for any $f\in L_2^{oe}({\mathbb {T}}^2)$, $\varepsilon _{\textbf{k}}\,f\in L_2^{oe}({\mathbb {T}}^2)$. This proves the invariance of the subspace $L_2^{oe}({\mathbb {T}}^2)$ with respect to the operator $H_0(\textbf{k})$.

By the condition (1.3) the kernel $v(p_1,p_2)$ of the integral operator V belongs to $L_2^{ee}({\mathbb {T}}^2).$ Hence, for $f\in L_2^{oe}({\mathbb {T}}^2)$, $g=Vf$ belongs to the subspace $L_2^{oe}({\mathbb {T}}^2)$, where

$$\begin{aligned} (Vf)(p_1,p_2)=\frac{1}{2\pi }\int \limits _{{\mathbb {T}}^2} v (p_1-s_1,p_2-s_2)f(s_1,s_2)\, \,\textrm{d}s_1 \,\textrm{d}s_2. \end{aligned}$$

By the above relations, we can conclude that $L_2^{\textrm{oe}}(\mathbb {T}^2)$ is invariant with respect to the operator $H(\textbf{k}) = H_0(\textbf{k}) - V$.

Since the operator $H(\textbf{k})$ is self-adjoint, the orthogonal complement $L_2^{eo}({\mathbb {T}}^2)={(L_2^{oe}({\mathbb {T}}^2))}^\bot$ is invariant with respect to the operator $H(\textbf{k})$.

Using the equalities (1.3), we obtain the explicit forms for the operators $V^{oe}=V\big |_{L_2^{oe}({\mathbb {T}}^2)}$ and $V^{eo}=V\big |_{L_2^{eo}({\mathbb {T}}^2)}$ as

$$(V^{oe}f)(\textbf{p}) =\frac{1}{(2\pi )^2}\int _{{\mathbb {T}}^2}\Big \{ 2\bar{v}(1)\sin p_1 \sin q_1+2\bar{v}(2)[2\sin p_1 \cos p_2\sin q_1 \cos q_2+$$

$$+\sin 2p_1 \sin 2q_1 ]+2 \sum _{n=2}^\infty \bar{v}(n) \, [ 2 \sin p_1 \cos (n-1)p_2 \sin q_1 \cos (n-1)q_2+$$

$$\begin{aligned} +2 \sin 2p_1 \cos (n-2)p_2 \sin 2q_1 \cos (n-2)q_2 ]\Big \}f(\textbf{q})\,\textrm{d}\textbf{q}, \;\;f\in L_2^{oe}({\mathbb {T}}^2); \end{aligned}$$

$$(V^{eo}f)(\textbf{p}) =\frac{1}{(2\pi )^2}\int _{{\mathbb {T}}^2}\Big \{ 2\bar{v}(1)\sin p_2 \sin q_2+2\bar{v}(2)[\sin 2p_2 \sin 2q_2+$$

$$+2\cos p_1 \sin p_2\cos q_1 \sin q_2 ]\,+$$

$$+\sum _{n=3}^\infty \bar{v}(n) [\,\sin n p_2 \sin n q_2 + 2 \cos p_1 \,\sin (n-1) p_2 \,\cos q_1 \,\sin (n-1) q_2+$$

$$\begin{aligned} +2 \cos 2p_1 \,\sin (n-2) p_2 \,\cos 2q_1 \,\sin (n-2) q_2] \, \Big \}f(\textbf{q})\,\textrm{d}\textbf{q}, \;\;f\in L_2^{eo}({\mathbb {T}}^2). \end{aligned}$$

Let us choose the system $\psi ^e_0(q)=\dfrac{1}{\sqrt{2\pi }}, \,\,\, \big \{\psi ^e_n(q)=\dfrac{1}{\sqrt{\pi }}{\cos {n q}} \,\big \}_{ n\in \mathbb {N}}\,\,\,$ as an orthonormal basis of the space $L_2^{e}({\mathbb {T}})$, and the system $\big \{\psi ^o_n(q)=\dfrac{1}{\sqrt{\pi }}{\sin {n q}}\big \}_{ n\in \mathbb {N}}$ as an orthonormal basis of the space $L_2^{o}({\mathbb {T}})$. Let $L^{e}(n-1)$ and $L^{o}(n)$ be the one-dimensional subspace spanned by the vector $\psi ^e_{n-1},$ and $\psi ^o_n, \,\,\,\,{ n\in \mathbb {N}},$ respectively. The spaces $L_2^{e}({\mathbb {T}})$ and $L_2^{o}({\mathbb {T}})$ are decomposed into the direct sums:

$$\begin{aligned} L_2^e({\mathbb {T}})=\sum _{n=1}^\infty \oplus L^{e}(n-1) \quad \text{ and } \quad L_2^o({\mathbb {T}})=\sum _{n=1}^\infty \oplus L^{o}(n). \end{aligned}$$

It gives the following decompositions

$$L_2^{o}({\mathbb {T}})\otimes L_2^{e}({\mathbb {T}})=\sum _{n=1}^\infty \oplus \big \{ L_2^{o}({\mathbb {T}})\otimes L^{e}(n-1) \big \}=\sum _{n=1}^\infty \oplus \mathfrak {B}_{n}^{oe},$$

$$L_2^{e}({\mathbb {T}})\otimes L_2^{o}({\mathbb {T}})=\sum _{n=1}^\infty \oplus \{ L_2^{e}({\mathbb {T}})\otimes L^{o}(n) \}=\sum _{n=1}^\infty \oplus \mathfrak {B}_{n}^{eo},$$

where $\mathfrak {B}_{n}^{oe}:=L_2^{o}({\mathbb {T}})\otimes L^{e}(n-1)\quad \text{ and } \quad \mathfrak {B}_{n}^{eo}:=L_2^{e}({\mathbb {T}})\otimes L^{o}(n).$

Lemma 3.2

For any ${ n\in \mathbb {N}}$ the subspace $\mathfrak {B}_{n}^{oe}$ is invariant under the operator $H^{oe}(k_1,\pi ).$

Proof

Let $f\,\psi ^e_{n-1}\in \mathfrak {B}_{n}^{oe}$. Then, it is represented as

$$\big (f\,\psi ^e_{n-1}\big )(p_1,p_2):= f(p_1)\,\psi ^e_{n-1}(p_2),$$

where $f\in L_2^{o}({\mathbb {T}}),$ $\psi _{n-1}^e\in L^{e}( n-1)$. Hence, the operator $H^{oe}(k_1,\pi )$ acts in $L_2^{oe}({\mathbb {T}}^2)$ as

$$H^{oe}(k_1,\pi )f\,\psi ^e_{n-1}=H_0(k_1,\pi )f\,\psi ^e_{n-1}-V^{oe}f\,\psi ^e_{n-1},$$

where

$$\begin{aligned} (H_0(k_1,\pi )f \psi ^e_{n-1})(p_1,p_2)=\Big [(4-2\cos \frac{k_1}{2}\cos p_1)f(p_1) \Big ]\psi ^e_{n-1}(p_2), \end{aligned}$$

(3.1)

$$\begin{aligned} (V^{oe}f\psi ^e_{n-1})(p_1,p_2)=\frac{1}{(2\pi )^2}\int _{{\mathbb {T}}^2}\Big \{ 2 \sum _{m=2}^\infty \bar{v}(m) \, [ 2 \sin p_1 \cos (m-1)p_2 \sin q_1 \cos (m-1)q_2 + \end{aligned}$$

$$\begin{aligned}\,+2 \sin 2p_1 \cos (m-2)p_2 \sin 2q_1 \cos (m-2)q_2 ]\Big \}f(q_1)\psi ^e_{n-1}(q_2)\, dq_1 dq_2= \end{aligned}$$

$$\begin{aligned} =\Big [\frac{1}{\pi } \int \limits _{{\mathbb {T}}}\{\bar{v}(n)\sin p_1 \sin q_1+\bar{v}(n+1)\sin 2p_1 \sin 2q_1 \}f(q_1)d q_1\Big ]\psi ^e_{n-1}(p_2). \end{aligned}$$

(3.2)

When obtaining equality (3.2), we used the orthogonality of the system $\{{\psi ^e_{n-1}}\}_{ n\in \mathbb {N}}$. By Eqs. 3.1 and 3.2, we get

$$(H^{oe}(k_1,\pi )f\psi ^e_{n-1})(p_1,p_2)=(H_0(k_1,\pi )f \psi ^e_{n-1})(p_1,p_2)-(V^{oe}f\psi ^e_{n-1})(p_1,p_2)=$$

$$=\Big [(4-2\cos \frac{k_1}{2}\cos p_1)f(p_1)-$$

$$-\frac{1}{\pi } \int \limits _{{\mathbb {T}}}\{\bar{v}(n)\sin p_1 \sin q_1+\bar{v}(n+1)\sin 2p_1 \sin 2q_1 \}f(q_1)d q_1 \Big ]\psi ^e_{n-1}(p_2) \in \mathfrak {B}_{n}^{oe},$$

which proves the Lemma.

The expressions Eqs. 3.1 and 3.2 show that the restriction $H^{oe}_n(k_1,\pi )$ of the operator $H^{oe}(k_1,\pi )$ to the subspace $\mathfrak {B}_{n}^{oe}:=L_2^{o}({\mathbb {T}})\otimes L^{e}(n-1)$ has the form

$$\begin{aligned} H^{oe}_n(k_1,\pi )= [H_0(k_1)- V_n^o]\otimes I_n, \end{aligned}$$

(3.3)

where $I_n$ is the unit operator in $L^{e}(n-1),$ and $H^{o}_n(k_1):= H_0(k_1)- V_n^o$ is the discrete Schrödinger operator acting on $L_2^{o}({\mathbb {T}})$ by the form

$$\begin{aligned} (H^{o}_n(k_1)f)(p)=\varepsilon _{k_1}(p)f(p)-\frac{1}{\pi } \int \limits _{{\mathbb {T}}}[\bar{v}(n)\sin p \sin q+\bar{v}(n+1)\sin 2p \sin 2q ]f(q)dq, \end{aligned}$$

(3.4)

where $\varepsilon _{k_1}(p)=4-2\cos \dfrac{k_1}{2}\cos p.$

Lemma 3.3

For any $n\in {\mathbb {N}}$ the subspace $\mathfrak {B}_{n}^{eo}$ is invariant under the operator $H^{eo}(k_1,\pi )$.

The Lemma 3.3 is proved in a similar way to the proof of Lemma 3.3.

We consider the restriction $H^{eo}_n(k_1,\pi )$ of the operator $H^{eo}(k_1,\pi )$ to the invariant subspace $\mathfrak {B}_{n}^{eo}=L_2^{e}({\mathbb {T}}) \otimes L^{o}(n):$

$$\begin{aligned} H^{eo}_n(k_1,\pi )=[H_0(k_1)-V_n^e]\otimes I_n, \end{aligned}$$

(3.5)

where $I_n$ is the unit operator on $L^{o}(n)$ and the discrete Schrödinger operator $H^{e}_n(k_1 )=H_0(k_1)-V_n^e$ acting on $L_2^{e}({\mathbb {T}})$ as

$$(H^{e}_n(k_1)f)(p)=\varepsilon _{k_1}(p)f(p)-$$

$$\begin{aligned} -\frac{1}{2\pi } \int \limits _{{\mathbb {T}}}[\bar{v}(n)+2\bar{v}(n+1)\cos p \cos q +2\bar{v}(n+2)\cos 2p \cos 2q]f(q)dq.\end{aligned}$$

(3.6)

The operators $H^{oe}(k_1, \pi )$ and $H^{eo}(k_1, \pi )$ are decomposed into direct sums:

$$\begin{aligned} H^{oe}(k_1, \pi )=\sum _{n=1}^{\infty }\oplus H^{oe}_n(k_1, \pi ), \qquad H^{eo}(k_1, \pi )=\sum _{n=1}^{\infty }\oplus H^{eo}_n(k_1, \pi ). \end{aligned}$$

(3.7)

Here the operators $H^{oe}_n(k_1,\pi )$ and $H^{eo}_n(k_1,\pi )$ are defined by Eqs. 3.3 and 3.5, respectively. Note that for any $n \in \mathbb {N}$ the essential spectra of the operators $H^{oe}_n(k_1,\pi )$ and $H^{eo}_n(k_1,\pi )$ coincide.

On the spectrum of the operator $H^{oe}(k_1,\pi )$

By the representation (3.3) it is sufficient to study which the eigenvalues of the Schrödinger operators $H^{o}_n(k_1 ), n\in {\mathbb {N}}$ defined by the form (3.4).

Note that the width $w(k_1):=w(k_1,\pi )$ of the essential spectrum of the operator $H^{o}_n(k_1 )$ does not depend on n and

$$\begin{aligned} w(k_1)=w_1(k_1)=4\cos \frac{k_1}{2}. \end{aligned}$$

(4.1)

It is known that the study of eigenvalues of the operator $H^{o}_n(k_1 )$, lying below the essential spectrum, leads to the study of eigenvalues of the self-adjoint compact positive operator $T^o_n(k_1,z)=r_0^{\frac{1}{2}}(k_1,z)V_n^o r_0^{\frac{1}{2}}(k_1,z)$, where $r_0(k_1,z)$ is the resolvent of the unperturbed operator $H_0(k_1 )$ (see [21]).

Denote by $m(k_1)$ and $M(k_1)$ the minimum and maximum values of the function $\varepsilon _{k_1}.$

For any $z \in \left( -\infty , m(k_1)\right)$ the operator $T^o_n(k_1,z)$ acts in $g \in L_2^{o}({\mathbb {T}})$ by the form

$$\begin{aligned} (T^o_n(k_1,z) g )(p)=\dfrac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\big [\bar{v}(n)\sin p \sin s +\bar{v}(n+1)\sin 2p \sin 2s \big ]g(s)ds}{\sqrt{\varepsilon _{k_1}(p)-z}\sqrt{\varepsilon _{k_1}(s)-z}}. \end{aligned}$$

For any $n\in {\mathbb {N}}$ the operator $T^o_n(k_1,z)$ is the operator of rank two.

Lemma 4.1

The number $z \in \left( -\infty , m(k_1)\right) ,$ is an eigenvalue of the operator $H^{o}_n(k_1 )$ if and only if 1 is the eigenvalue of the operator $T^o _n(k_1,z).$

Lemma 4.2

(Birman-Schwinger principle). The number of eigenvalues of the operator $H^{o}_n(k_1 )$ lying below z,  $z \in \left( -\infty , m(k_1)\right) ,$ is equal to the number of eigenvalues of the operator $T^o_n(k_1,z)$ greater than one, i.e.,

$$\begin{aligned} n\bigg [1,T^o_n\left( k_1,z\right) \bigg ]=n\bigg [-z,-H^{o}_n(k_1)\bigg ]. \end{aligned}$$

The proofs of Lemmas 4.1 and 4.2 are similar to the proofs of Lemmas 3.1 and Theorem 3.2 of [21].

For any $n \in {\mathbb {N}}$, we introduce the following notations

$$\begin{aligned} C_1(n)={\bar{v}(n)+2\bar{v}(n+1)}+\sqrt{\bar{v}^2(n)+4\bar{v}^2(n+1)}=\frac{2c_0}{e^{n+1}}p^+, \end{aligned}$$

(4.2)

$$\begin{aligned} C_2(n)={\bar{v}(n)+2\bar{v}(n+1)}-\sqrt{\bar{v}^2(n)+4\bar{v}^2(n+1)}=\frac{2c_0}{e^{n+1}}p^-, \end{aligned}$$

(4.3)

where

$$\begin{aligned} p^+=\dfrac{e+2+\sqrt{e^2+4}}{2}\approx 4.0465, \quad p^-=\dfrac{e+2-\sqrt{e^2+4}}{2}\approx 0.671. \end{aligned}$$

Since $p^+>p^-,$ the inequality $C_1(n)>C_2(n)$ holds for any $n \in {\mathbb {N}}$.

Theorem 4.3

Let $n \in {\mathbb {N}}$ and $k_1 \in (-\pi , \pi ].$

1) If

$$\begin{aligned} C_1(n)< \dfrac{w(k_1)}{2}, \end{aligned}$$

then the operator $H^{o}_n(k_1 )$ has no eigenvalues lying below the essential spectrum;

2) if

$$\begin{aligned} C_1(n)< \dfrac{w(k_1)}{2}<C_2(n), \end{aligned}$$

then the operator $H^{o}_n(k_1 )$ has a unique eigenvalue lying below the essential spectrum;

3) if

$$\begin{aligned} C_2(n)> \dfrac{w(k_1)}{2}, \end{aligned}$$

then the operator $H^{o}_n(k_1 )$ has two eigenvalues lying below the essential spectrum.

Proof

Since the function $\varepsilon _{k_1}$ has a non-degenerate minimum at the origin and from the representation $T^o_n(k_1,z)$ it follows that the limit operator $T^o_n(k_1,m(k_1)):=T^o_n(k_1)$ is a bounded self-adjoint operator on $L_2^o(\mathbb {T})$.

Let $g \in L_2^o(\mathbb {T})$ be a non-zero solution of the equation

$$\begin{aligned} (T^o_n(k_1)g)(p)=\lambda g(p). \end{aligned}$$

This equation gives

$$\begin{aligned} g(p)=\dfrac{1}{\lambda \sqrt{\varepsilon _{k_1}(p)-m(k_1)}}\bigg [a_1\bar{v}(n)\sin p +a_2 \bar{v}(n+1)\sin 2p \bigg ], \end{aligned}$$

(4.4)

where

$$\begin{aligned} a_1=\frac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin s g(s)ds }{\sqrt{\varepsilon _{k_1}(s)- m(k_1)}}, \quad a_2=\frac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin 2s g(s)ds }{\sqrt{\varepsilon _{k_1}(s) -m(k_1)}}. \end{aligned}$$

(4.5)

Substituting Eqs. 4.4 into 4.5, we obtain a homogeneous system of equations for $a_1$ and $a_2:$

$$\begin{aligned} {\left\{ \begin{array}{ll} \begin{aligned} \lambda a_1=\,\,\dfrac{\bar{v}(n)}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin ^2 s ds }{{\varepsilon _{k_1}(s)-m(k_1)}} a_1+ \,\dfrac{\bar{v}(n+1)}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin s \sin 2s ds }{{\varepsilon _{k_1}(s)-m(k_1)}} a_2, \\ \lambda a_2=\,\,\dfrac{\bar{v}(n)}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin s \sin 2s ds }{{\varepsilon _{k_1}(s)-m(k_1)}} a_1+ \,\dfrac{\bar{v}(n+1)}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{ \sin ^2 2s ds }{{\varepsilon _{k_1}(s)-m(k_1)}} a_2. \end{aligned} \end{array}\right. } \end{aligned}$$

(4.6)

The system of Eq. 4.6 has a non-zero solution if and only if the determinant is equal to zero, i.e.,

$$\begin{aligned} & \Delta (\lambda , k_1) = \left| \begin{array}{ccc} \lambda - \bar{v}(n)\,\Delta _{11}(k_1) & & -{ \bar{v}(n+1)}\Delta _{12}(k_1) \\ & & \\ -{\bar{v}(n)}\Delta _{21}(k_1) & & \lambda - \bar{v}(n+1) \Delta _{22}(k_1) \end{array}\right| =0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \nonumber \end{aligned}$$

(4.7)

where

$$\begin{aligned} \Delta _{11}(k_1)=\dfrac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin ^2 s ds }{{\varepsilon _{k_1}(s)-m(k_1)}}, \end{aligned}$$

(4.8)

$$\begin{aligned} \Delta _{12}(k_1)=\Delta _{21}(k_1)=\dfrac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin s \sin 2s ds }{{\varepsilon _{k_1}(s)-m(k_1)}}, \end{aligned}$$

(4.9)

$$\begin{aligned} \Delta _{22}(k_1)=\dfrac{1}{\pi }\int \limits _{{\mathbb {T}}}\dfrac{\sin ^2 2s ds }{{\varepsilon _{k_1}(s)-m(k_1)}}. \end{aligned}$$

(4.10)

Taking into account the equalities $\,\,\,\,\varepsilon _{k_1}(p)-m(k_1)=2\cos \dfrac{k_1}{2}(1-\cos p)=w(k_1)\sin ^2 \dfrac{p}{2}$ and calculating the integrals Eqs. 4.8, 4.9 and 4.10, we get

$$\Delta _{11}(k_1)= \Delta _{12}(k_1)=\dfrac{1}{\cos \dfrac{k_1}{2}}=\dfrac{4}{w(k_1)}, \quad \Delta _{22}(k_1)= \dfrac{2}{\cos \dfrac{k_1}{2}}=\dfrac{8}{w(k_1)}.$$

Substituting the above equalities into Eq. 4.7, we obtain:

$$\Delta (\lambda , k_1)=\left( \lambda -\dfrac{\bar{v}(n)}{\cos \dfrac{k_1}{2}}\right) \left( \lambda -2\dfrac{\bar{v}(n+1)}{\cos \dfrac{k_1}{2}}\right) -\dfrac{\bar{v}(n)\bar{v}(n+1)}{\cos ^2\dfrac{k_1}{2}}=$$

$$=\lambda ^2-\left( \dfrac{\bar{v}(n)+2\bar{v}(n+1)}{\cos ^2\dfrac{k_1}{2}}\right) \lambda +\dfrac{\bar{v}(n)\bar{v}(n+1)}{\cos ^2\dfrac{k_1}{2}}=0.$$

Having found the roots $\lambda _1(n, k_1)$ and $\lambda _2(n, k_1)$ of this equation, expressing them through $C_1(n)$, $C_2(n)$ and $w(k_1)$, we obtain the following equations (see the forms (4.2), (4.3) and (4.1)):

$$\lambda _1(n, k_1)=\dfrac{2C_1(n)}{w(k_1)}, \quad \lambda _2(n, k_1)=\dfrac{2C_2(n)}{w(k_1)}.$$

The statements of the Theorem directly follow from this and from the Lemma 4.2.

For any $\beta \in (0,\pi )$, we introduce the following notation:

$$\begin{aligned} card \left\{ n\in {\mathbb {N}}: C_1(n)>2\sin \beta \right\} =n_1(\beta ), \end{aligned}$$

(4.11)

$$\begin{aligned} card \left\{ n\in {\mathbb {N}}: C_2(n)>2\sin \beta \right\} =n_2(\beta ). \end{aligned}$$

(4.12)

Note that the equality

$$\begin{aligned} N(\beta )=n_1(\beta )+n_2(\beta ) \end{aligned}$$

(4.13)

holds.

Theorem 4.4

Let $\beta \in (0, \pi ).$ 1) If the inequality $C_1(1)<2\sin \beta$ holds, then the operator $H^{oe}(\textbf{k}_{\beta })$ has no eigenvalues lying below the essential spectrum.

2) For any $\beta \in (0,\pi )$ the operator $H^{oe}(\textbf{k}_{\beta })$ has exactly $N(\beta )=n_1(\beta )+n_2(\beta )$ eigenvalues (with the multiplicity) lying below the essential spectrum.

Proof

It follows directly from the decomposition (3.7) and the proof of Theorem 4.3.

Proof of the Theorem 1.2

From the definition of the functions $C_1(n)$ and $C_2(n)$, it follows that they form a geometric progression with the denominator $e^{-1}$ and it can be easily verified that the following inequalities hold:

$$C_1(n+1)>C_2(n)>C_1(n+2)>C_2(n+1), \quad n \in \mathbb {N}.$$

In addition, $C_1(1)>C_1(2)$, thus from Eqs. 4.11 and 4.12, we have:

$$\begin{aligned} n_2(\beta )\ge n_1(\beta )-2. \end{aligned}$$

(4.14)

Solutions

$$\begin{aligned} \beta _n^+=\arcsin \dfrac{c_0p^+}{e^{n+1}}, \quad \beta _n^-=\arcsin \dfrac{c_0p^-}{e^{n+1}} \end{aligned}$$

(4.15)

of the equations $C_1(n)=2\sin \beta$ and $C_2(n)=2\sin \beta$ are the jump points of the functions $n_1(\beta )$ and $n_2(\beta )$, respectively (see Theorem 4.3).

Let

$$\begin{aligned} n_1(\beta )=n. \end{aligned}$$

(4.16)

The definition of $n_1(\beta )$ implies that, the Eq. 4.16 is equivalent to the relation

$$\begin{aligned} C_1(n+1)<2\sin \beta <C_1(n). \end{aligned}$$

(4.17)

We rewrite (4.17) as

$$\dfrac{c_0p^+}{e^{n+2}}<\sin \beta <\dfrac{c_0p^+}{e^{n+1}}.$$

From here, we have

$$\log (c_0p^+)-n-2<\log \sin \beta <\log (c_0p^+)-n-1.$$

Hence, denoting by $d=\log (c_0p^+)-1,$ we get

$$n<d-\log {\sin \beta }<n+1.$$

From here and Eq. 4.16, we get

$$1<\dfrac{d}{n_1(\beta )}-\dfrac{\log \sin \beta }{n_1(\beta )}<1+\frac{1}{n_1(\beta )}.$$

And from this follows

$$\begin{aligned} \lim _{\beta \rightarrow 0}\dfrac{n_1(\beta )}{|\log \sin \beta |}=1. \end{aligned}$$

(4.18)

Since $n_2(\beta )\le n_1(\beta ),$ it follows

$$2n_2(\beta )\le n_2(\beta )+ n_1(\beta )\le 2n_1(\beta ).$$

By the relations Eqs. 4.14 and 4.13, we get

$$2n_1(\beta )-4\le N(\beta )\le 2n_1(\beta ).$$

It follows

$$\begin{aligned} \lim _{\beta \rightarrow 0}\dfrac{N(\beta )}{n_1(\beta )} =2. \end{aligned}$$

(4.19)

Using the limit property for Eqs. 4.18 and 4.19, we obtain (1.4).   

Further, we study the eigenvalues of the operator $H^{o}_n(\pi -2\beta )$ for small $\beta$. From the representation (3.4) it follows that for a fixed $n\in {\mathbb {N}}$ the operator $H^{o}_n(\pi )$ has two simple eigenvalues

$$z_{n}(\pi )=4-\bar{v}(n), \quad z_{n+1}(\pi )=4-\bar{v}(n+1).$$

Theorem 4.5

For every $n\in {\mathbb {N}}$ there exists $\delta _n>0$ such that for any $\beta \in (0,\delta _n)$ the operator $H^{o}_n(\pi -2\beta )$ has two different simple eigenvalues $\,\,z_{n}^{(1)o}(\beta )$ and $\,\,\,z_{n+1}^{(2)o}(\beta )$ with the following asymptotics

$$\begin{aligned} {\begin{matrix} z_{n}^{(1)o}\left( \beta \right) =z_{n}(\pi )-\frac{e^{n+1}}{c_0(e-1)}\beta ^2+O(\beta ^4), \,\,\,\, \beta \rightarrow 0, \\ z_{n+1}^{(2)o}\left( \beta \right) =z_{n+1}(\pi )- \dfrac{e^{n+1}(e-2)}{c_0(e-1)}\beta ^2+O(\beta ^4), \,\,\, \beta \rightarrow 0. \end{matrix}} \end{aligned}$$

The proof of Theorem 4.5 is similar to the proof of Theorem 6.1 in [22].

From Theorems 4.4 and 4.5, we directly obtain the following theorem.

Theorem 4.6

a) Operator $H^{oe}(\textbf{k}_{\beta } )$ has simple eigenvalue lying in a some neighborhood of the double eigenvalue $z_1(\varvec{\pi })=4-\bar{v}(1)$ of the operator $H(\varvec{\pi }):$

$$\begin{aligned} z_{1}^{(1)o}(\beta )=z_1(\varvec{\pi })-\frac{e^2}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0; \end{aligned}$$

b) for each $n\ge 2$ there exists $\delta _n>0$ such that for any $\beta \in (0, \delta _n)$ the operator $H^{oe}(\textbf{k}_{\beta })$ has two different simple eigenvalues lying in a some neighborhood of the fourfold eigenvalue $z_n(\varvec{\pi })=4-\bar{v}(n)$ of the operator $H(\varvec{\pi })$:

$$\begin{aligned} z_{n}^{(1)o}(\beta )=z_n(\varvec{\pi })-\frac{e^{n+1}}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned} z_{n}^{(2)o}(\beta )=z_n(\varvec{\pi })-\dfrac{e^n(e-2)}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0. \end{aligned}$$

On the spectrum of the operator $H^{eo}(k_1,\pi )$

By the representation (3.3) it is sufficient to study the eigenvalues of the Schrödinger operator $H^{e}_n(k_1 ), n\in {\mathbb {N}}$ defined by form (3.6).

The width $w(k_1):=w(k_1,\pi )$ of the essential spectrum of the operator $H^{e}_n(k_1 )$ coincides with Eq. 4.1.

It is known that the study of eigenvalues of the operator $H^{e}_n(k_1 )$ lying below the essential spectrum, leads to the study of eigenvalues of the self-adjoint compact positive operator $T^e_n(k_1,z)=r_0^{\frac{1}{2}}(k_1,z)V_n^e r_0^{\frac{1}{2}}(k_1,z).$

For any $z \in \left( -\infty , m(k_1)\right)$ the operator $T^e_n(k_1,z)$ acts in $g \in L_2^{e}({\mathbb {T}})$ by the formula

$$\begin{aligned} (T^e_n(k_1,z) g )(p)=\dfrac{1}{2\pi }\int \limits _{{\mathbb {T}}}\dfrac{\big [\bar{v}(n)+2\bar{v}(n+1)\cos p \cos q +2\bar{v}(n+2)\cos 2p \cos 2q \big ]g(s)ds}{\sqrt{\varepsilon _{k_1}(p)-z}\sqrt{\varepsilon _{k_1}(s)-z}}. \end{aligned}$$

Note that for any $n\in {\mathbb {N}}$ the operator $T^e_n(k_1,z)$ is the operator of rank three.

Lemma 5.1

The number $z<m(k_1)$ is an eigenvalue of the operator $H^{e}_n(k_1 )$ if and only if 1 is the eigenvalue of the operator $T^e _n(k_1,z).$

Lemma 5.2

The number of eigenvalues of the operator $H^{e}_n(k_1 )$ lying below $z< m(k_1)$ is equal to the number of eigenvalues of the operator $T^e_n(k_1,z)$ greater than unity, i.e.,

$$\begin{aligned} n\bigg [1,T^e_n\left( k_1,z\right) \bigg ]=n\bigg [-z,-H^{e}_n(k_1)\bigg ]. \end{aligned}$$

Theorem 5.3

For any $n\in {\mathbb {N}}$ the operator $H^{e}_n(k_1 )$ has at least one eigenvalue lying below the essential spectrum.

Proof

According to the Lemma 5.2, we show that the operator $T^e_n(k_1,z)$ has at least one eigenvalue greater than 1 for some $z_0<m(k_1)$.

Let $g_0(q)=\dfrac{\sqrt{C_0(z)}}{{\sqrt{2\pi }}\sqrt{\varepsilon _{k_1}(q)-z}} \in L_2^{e}({\mathbb {T}})$ and $\Vert g_0\Vert =1,$ where $C_0(z)$ is the normalizing factor. Note that for $z<m(k_1)$ the following assertions hold:

$$\begin{aligned} C_0(z)={\sqrt{(4-z)^2-4\cos ^2 \dfrac{k_1}{2}}}=\left( \dfrac{1}{2\pi }{\int \limits _{{\mathbb {T}}}} \dfrac{dq}{\varepsilon _{k_1}(q)-z}\right) ^{-1}\end{aligned}$$

(5.1)

and

$$\begin{aligned} \lim \limits _{z\rightarrow m(k_1)}C_0(z)=0. \end{aligned}$$

(5.2)

For the scalar product $(T^{e}_n(k_1,z )g_0,g_0)$, we obtain

$$(T^{e}_n(k_1,z )g_0,g_ 0)=\frac{1}{2\pi }\int \limits _{{\mathbb {T}}}\int \limits _{{\mathbb {T}}} T^e_n\left( p,q;k_1,z\right) g_0(q)dq\overline{g_0(p)}dp=$$

$$=\frac{\bar{v}(n)}{2\pi }{\left| {\int \limits _{{\mathbb {T}}}}\frac{g_0(q)dq}{\sqrt{\varepsilon _{k_1}(q)-z}}\right| }^2+ \frac{\bar{v}(n+1)}{\pi }{\left| {\int \limits _{{\mathbb {T}}}}\frac{\cos q g_0(q)dq}{\sqrt{\varepsilon _{k_1}(q)-z}}\right| }^2 +\frac{\bar{v}(n+2)}{\pi }{\left| {\int \limits _{{\mathbb {T}}}}\frac{\cos 2q g_0(q)dq}{\sqrt{\varepsilon _{k_1}(q)-z}}\right| }^2.$$

All the addends of this equation are non-negative, so according to Eq. 5.1, we have

$$\begin{aligned} (T^{e}_n(k_1,z )g_0,g_ 0)\ge \frac{\bar{v}(n)C_0(z)}{(2\pi )^2}{\int \limits _{{\mathbb {T}}}}\frac{dp}{\varepsilon _{k_1}(p)-z}{\int \limits _{{\mathbb {T}}}}\frac{dq}{{\varepsilon _{k_1}(q)-z}}= \frac{\bar{v}(n)}{C_0(z)}. \end{aligned}$$

From this and Eq. 5.2, there exists a number $z=z_0(n)<m(k_1)$ such that for all $z\in \left( z_0(n), m(k_1)\right)$, we have $\dfrac{\bar{v}(n)}{C_0(z)}>1$. Then, from the Birman-Schwinger principle (see Lemma 5.2) it follows that $n\big [-m(k_1), -H^e_n(k_1)\big ]\ge 1$.

Proof of the Theorem 1.3

It follows from the proof of Theorem 5.3 and equality (3.7).

Let us present statements without proof regarding the eigenvalues of the operator $H^{e}_n(\pi -2\beta )$ for small $\beta$. Note that for a fixed $n\in {\mathbb {N}}$ the operator $H^{e}_n(\pi )$ has three simple eigenvalues

$$z_n(\pi )=4-\bar{v}(n), \quad z_{n+1}(\pi )=4-\bar{v}(n+1), \quad z_{n+2}(\pi )=4-\bar{v}(n+2).$$

Theorem 5.4

For each $n\in {\mathbb {N}}$ there exists $\delta _n>0$ such that for any $\beta \in (0,\delta _n)$ the operator $H^{e}_n(\pi -2\beta )$ has three eigenvalues $z_{n}^{(1)e}(\beta ), \,\,\,z_{n+1}^{(2)e}(\beta )$ and $\,\,\,z_{n+2}^{(3)e}(\beta )$ with the following asymptotics

$$\begin{aligned} z_{n}^{(1)e}(\beta )=z_n(\pi )-\frac{2e^{n+1}}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\, \beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned}z_{n+1}^{(2)e}(\beta )=z_{n+1}(\pi )- \frac{e^{n+1}(e-2)}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\, \beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned} z_{n+2}^{(3)e}(\beta )=z_{n+2}(\pi )-\dfrac{e^{n+2}(e-2)}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\, \beta \rightarrow 0. \end{aligned}$$

Therefore, according to the equality $H^{eo}(\textbf{k}_{\beta })=\sum _{n=1}^{\infty }\oplus H^{eo}_n(\textbf{k}_{\beta })$ the following statements are true.

Theorem 5.5

a) Operator $H^{eo}(\textbf{k}_{\beta })$ has a simple eigenvalue lying in a some neighborhood of the double eigenvalue $z_1(\varvec{\pi })=4-\bar{v}(1)$ of the operator $H(\varvec{\pi }):$

$$\begin{aligned} z_{1}^{(1)e}(\beta )=z_1(\varvec{\pi })-\frac{2e^2}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0; \end{aligned}$$

b) operator $H^{eo}(\textbf{k}_{\beta })$ has exactly two different simple eigenvalues lying in a some neighborhood of the fourfold eigenvalue $z_2(\varvec{\pi })=4-\bar{v}(2)$ of the operator $H(\varvec{\pi })$:

$$\begin{aligned} z_{2}^{(1)e}(\beta )=z_2(\varvec{\pi })-\frac{2e^3}{c_0(e-1)}\beta ^2+O(\beta ^4), \,\,\,\,\beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned} z_{2}^{(2)e}(\beta )=z_2(\varvec{\pi })-\dfrac{e^2(e-2)}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0; \end{aligned}$$

c) for each $n\ge 3$ there exists $\delta _n>0$ such that for any $\beta \in (0, \delta _n)$ the operator $H^{eo}(\textbf{k}_{\beta })$ has three eigenvalues

$$\begin{aligned} z_{n}^{(1)e}(\beta )=z_n(\varvec{\pi })-\frac{2e^{n+1}}{c_0(e-1)}\beta ^2+O(\beta ^4), \,\,\,\,\beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned} z_{n}^{(2)e}(\beta )=z_n(\varvec{\pi })-\frac{e^n(e-2)}{c_0(e-1)}\beta ^2 +O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0, \end{aligned}$$

$$\begin{aligned} z_{n}^{(3)e}(\beta )=z_n(\varvec{\pi })-\frac{e^n(e-2)}{c_0(e-1)}\beta ^2+O\left( \beta ^4\right) , \,\,\,\,\beta \rightarrow 0, \end{aligned}$$

lying in a small neighborhood of the value $z_n(\varvec{\pi }) = 4-\bar{v}(n).$