J. I. Abdullaev, A. M. Toshturdiev^*
Samarkand State University, University Boulevard 15, Samarkand, Uzbekistan.
DOI: 10.4236/jmp.2021.121004   PDF    HTML   XML   484 Downloads   1,471 Views   Citations

Abstract

We consider a Hamiltonian of a system of two fermions on a three-dimensional lattice Z³ with special potential . The corresponding Shrödinger operator H(k) of the system has an invariant subspac L^-₁₂₃(T³) , where we study the eigenvalues and eigenfunctions of its restriction H^-₁₂₃(k). Moreover, there are shown that H^-₁₂₃(k₁, k₂, π) has also infinitely many invariant subspaces , where the eigenvalues and eigenfunctions of eigenvalue problem are explicitly found.

Keywords

Hamiltonian, Fermion, Bound State, Shrödinger Operator, Invariant Subspace, Total Quasi-Momentum, Eigenvalue, Birman-Schwinger Principle

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1. Introduction

The nature of bound states of two-particle cluster operators for small parameter values was first studied in detail by Minlos and Mamatov [1] and then in a more general setting by Minlos and Mogilner [2]. In [3], Howland showed that the Rellich theorem on perturbations of eigenvalues does not extend to the resonance theory. Studying bound states of a two-particle system Hamiltonian H on the d-dimensional lattice ${\mathbb{Z}}^d$ reduces to studying [2] [4] [5] [6] [7] the eigenvalues of a family of Shrödinger operators $H\left(k\right)\mathrm{,}k\in {\mathbb{T}}^d$, where $k$ is the total quasi-momentum of a system. Moreover, eigenfunctions of $H\left(k\right)$ are interpreted as bound states of the Hamiltonian H, and eigenvalues, as the bound state energies. The bound states of H of a system of two fermions on a one-dimensional lattice were studied in [4], a system of two bosons on a two-dimensional lattice was studied in [6], and perturbations of the eigenvalues of a two-particle Shrödinger operator on a one-dimensional lattice were studied in [8]. The finiteness of the number of eigenvalues of Shrödinger operator on a lattice was studied in the works [7] [9].

The discrete spectrum of the two-particle continuous Shrödinger operator

$h_{\lambda }=-\Delta +\lambda V$

was studied by many authors, with the conditions for the potential V formulated in its coordinate representation. 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 [10]. If $V\le 0$, then the number of negative eigenvalues $N\left(\lambda \right)$ is a nondecreasing function of $\lambda \in \left(\mathrm{0,}\infty \right)$, and each eigenvalue $z_n\left(\lambda \right)$ decreases on the half-axis $\left(\mathrm{0,}\infty \right)$. 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 (see [10] ) 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 [11] [12]. Regarding the second question, it turns out that the bound state disappears by being absorbed into the continuous spectrum and becomes a resonance [5].

Here, we consider bound states of the Hamiltonian $\hat{H}$ (see (1)) of a system of two fermions on the three-dimensional lattice ${\mathbb{Z}}^3$ with the special potential $\hat{v}$ (see (5)). In other words, we study the discrete spectrum of a family of the Shrödinger operators $H\left(k\right)$, $k=\left(k_1\mathrm{,}{k}_2\mathrm{,}{k}_3\right)\in {\mathbb{T}}^3$, (see (3)) corresponding to $\hat{H}$ in the invariant subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$.

Restriction of the operator $H\left(k\right)$ in the invariant subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$ is denoted by $H_{123}^{-}\left(k\right)$.

In the case $k=\stackrel{\to }{\pi }\mathrm{<mo>\; :\; </mo>}=\left(\pi \mathrm{,}\pi \mathrm{,}\pi \right)$, the operator $H\left(\stackrel{\to }{\pi }\right)$ has an infinite number of eigenvalues of the form $6-\hat{v}\left(n\right)\mathrm{,}n\in {\mathbb{Z}}^3$ and the essential spectrum consists of the single point 6. Here, the potential $\hat{v}$ is defined by (5) and $\bar{v}\mathrm{<mo>\; :\; </mo>}\mathbb{N}\to ℝ$ is a decreasing function on $\mathbb{N}$ and $\bar{v}\in {\mathcal{l}}_2\left(\mathbb{N}\right)$. These eigenvalues $z_n\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n\right)\mathrm{,}n\in \mathbb{N}$ are arranged in ascending order, $z_1\left(\stackrel{\to }{\pi }\right)<\cdots <z_n\left(\stackrel{\to }{\pi }\right)<\cdots$, and the smallest eigenvalue $z_1\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(1\right)$ is threefold, $z_2\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(2\right)$ is sevenfold, and the other eigenvalues $z_n\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n\right)\mathrm{,}n\ge 3$ are ninefold. All ninefold eigenvalues $z_n\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n\right)\mathrm{,}n\ge 3$ of the operator $H\left(\stackrel{\to }{\pi }\right)$ are simple eigenvalues for the operator $H_{123}^{-}\left(\stackrel{\to }{\pi }\right)$.

Further, we investigate eigenvalues and eigenfunctions of the restriction operator $H_{123}^{-}\left(k\right)$.

In the case $k=\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ the corresponding operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ has infinitely many invariant subspaces ${\Re }_{123}^{-}\left(n\right)\mathrm{<mo>\; :\; </mo>}=L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$. It is proved that the restriction $H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ of the operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ in the invariant subspace ${\Re }_{123}^{-}\left(n\right)$ has no more than one eigenvalue. If exists, it can be calculated explicitly. For every $\left(k_1\mathrm{,}{k}_2\right)\in {\left(-\pi \mathrm{,}\pi \right)}^2$ the operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ has only a finite number of eigenvalues.

For any perturbation $\beta >0$, the essential spectrum $\left\{6\right\}$ of $H\left(\stackrel{\to }{\pi }\right)$ becomes the essential spectrum ${\sigma }_{ess}\left(H\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)\right)=\left[6-2\sin \beta \mathrm{,6}+2\sin \beta \right]$. If the potential $\hat{v}$ is of the form (5), the Shrödinger equation $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)f=zf\mathrm{,}f\in {\Re }_{123}^{-}\left(n\right)$ can be exactly solved (see Theorem 1).

The Shrödinger equations $H\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)f=zf$ and $H\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)f=zf\mathrm{,}f\in {\Re }_{123}^{-}\left(n\right)$ with small $\beta$ are solved by using methods invariant subspaces and operator theory.

2. Description of the Hamiltonian and Expansion in a Direct Integral

The free Hamiltonian ${\hat{H}}_0$ of a system of two fermions on a three-dimensional lattice ${\mathbb{Z}}^3$ usually corresponds to a bounded self-adjoint operator acting in the Hilbert space ${\mathcal{l}}_2^{as}\left({\mathbb{Z}}^3\times {\mathbb{Z}}^3\right)\mathrm{<mo>\; :\; </mo>}=\left\{f\in {\mathcal{l}}_2\left({\mathbb{Z}}^3\times {\mathbb{Z}}^3\right)\mathrm{<mo>\; :\; </mo>}f\left(x\mathrm{,}y\right)=-f\left(y\mathrm{,}x\right)\right\}$ by the formula

${\hat{H}}_0=-\frac{1}{2m}{\Delta }_1-\frac{1}{2m}{\Delta }_2\mathrm{.}$

Here, m is the fermion mass, which we assume to be equal to unity in what follows, ${\Delta }_1=\Delta \otimes I$ and ${\Delta }_2=I\otimes \Delta$, where I is the identity operator, and the lattice Laplacian $\Delta$ is a difference operator that describes a translation of a particle from a side to a neighboring side,

$\left(\Delta \hat{\psi }\right)\left(x\right)={\displaystyle \underset{j=1}{\overset{3}{\sum }}}\left[\hat{\psi }\left(x+e_j\right)+\hat{\psi }\left(x-e_j\right)-2\hat{\psi }\left(x\right)\right]\mathrm{,}x\in {\mathbb{Z}}^3\mathrm{,}\hat{\psi }\in {\mathcal{l}}_2\left({\mathbb{Z}}^3\right)\mathrm{,}$

where $e_1=\left(\mathrm{1,0,0}\right)\mathrm{,}{e}_2=\left(\mathrm{0,1,0}\right)\mathrm{,}{e}_3=\left(\mathrm{0,0,1}\right)$ are unit vectors in ${\mathbb{Z}}^3$. The total Hamiltonian $\hat{H}$ acts in the Hilbert space ${\mathcal{l}}_2^{as}\left({\mathbb{Z}}^3\times {\mathbb{Z}}^3\right)$ and is the difference of the free Hamiltonian ${\hat{H}}_0$ and the interaction potential ${\hat{V}}_2$ of the two fermions (see [8] [13] ):

$\hat{H}={\hat{H}}_0-{\hat{V}}_2\mathrm{,}$ (1)

where

$\left({\hat{V}}_2\hat{\psi }\right)\left(x,y\right)=\hat{v}\left(x-y\right)\hat{\psi }\left(x,y\right)\mathrm{,}\hat{\psi }\in {\mathcal{l}}_2^{as}\left({\left({\mathbb{Z}}^3\right)}^2\right)\mathrm{<mo>\; :\; </mo>}={\mathcal{l}}_2^{as}\left({\mathbb{Z}}^3\times {\mathbb{Z}}^3\right)\mathrm{.}$

Hereafter, we assume that

$\hat{v}\in {\mathcal{l}}_2\left({\mathbb{Z}}^3\right)\text{and}\hat{v}\left(x\right)=\hat{v}\left(-x\right)\ge 0\text{for}\text{all}x\in {\mathbb{Z}}^3\mathrm{.}$ (2)

Under this condition, the Hamiltonian $\hat{H}$ is a bounded self-adjoint operator in ${\mathcal{l}}_2^{as}\left({\left({\mathbb{Z}}^3\right)}^2\right)$.

We pass to momentum representation using the Fourier transform [2] [4] [7]

$F\mathrm{<mo>\; :\; </mo>}{\mathcal{l}}_2^{as}\left({\mathbb{Z}}^3\times {\mathbb{Z}}^3\right)\to L_2^{as}\left({\mathbb{T}}^3\times {\mathbb{T}}^3\right)\mathrm{.}$

The Hamiltonian $H=H_0-V=F\hat{H}{F}^{-1}$ in the momentum representation commutes with the unitary operators $U_s\mathrm{,}s\in {\mathbb{Z}}^3$, given by

$\left(U_{s}f\right)\left(k_1\mathrm{,}{k}_2\right)=\exp \left(-i\left(s\mathrm{,}{k}_1+k_2\right)\right)f\left(k_1\mathrm{,}{k}_2\right)\mathrm{,}f\in L_2^{as}\left({\mathbb{T}}^3\times {\mathbb{T}}^3\right)\mathrm{.}$

It follows that there exist decompositions of $L_2^{as}\left({\mathbb{T}}^3\times {\mathbb{T}}^3\right)$ and the operators $U_s$ and H into direct integrals (see [7] [9] and [10] )

$L_2^{as}\left({\mathbb{T}}^3\times {\mathbb{T}}^3\right)={\displaystyle {\int }_{{\mathbb{T}}^3}}\oplus L_2^{as}\left(F_k\right)\text{d}k\mathrm{,}{U}_s={\displaystyle {\int }_{{\mathbb{T}}^3}}\oplus U_s\left(k\right)\text{d}k\mathrm{,}H={\displaystyle {\int }_{{\mathbb{T}}^3}}\oplus \tilde{H}\left(k\right)\text{d}k\mathrm{.}$

Here,

$F_k=\left\{\left(k_1\mathrm{,}{k}_2\right)\in {\mathbb{T}}^3\times {\mathbb{T}}^3\mathrm{<mo>\; :\; </mo>}{k}_1+k_2=k\right\}\mathrm{,}k\in {\mathbb{T}}^3\mathrm{,}$

and $U_s\left(k\right)$ is an operator of multiplication by the function $\exp \left(-i\left(s,k\right)\right)$ in $L_2^{as}\left(F_k\right)$. The fiber operator $\tilde{H}\left(k\right)$ of H also acts in $L_2^{as}\left(F_k\right)$ and is unitarly equivalent to $H\left(k\right)\mathrm{<mo>\; :\; </mo>}=H_0\left(k\right)-V$, which is called the Shrödinger operator. This operator acts in the Hilbert space $L_2^o\left({\mathbb{T}}^3\right)\mathrm{<mo>\; :\; </mo>}=\left\{f\in L_2\left({\mathbb{T}}^3\right)\mathrm{<mo>\; :\; </mo>}f\left(-q\right)=-f\left(q\right)\right\}$ by the formula

$\left(H\left(k\right)f\right)\left(q\right)={\epsilon }_k\left(q\right)f\left(q\right)-{\left(2\pi \right)}^{-\frac{3}{2}}{\displaystyle {\int }_{{\mathbb{T}}^3}}v\left(q-s\right)f\left(s\right)\text{d}s\mathrm{.}$ (3)

The unperturbed operator $H_0\left(k\right)$ is an operator of multiplication by the function

$\begin{array}{c}{\epsilon }_k\left(q\right)=\epsilon \left(\frac{k}{2}+q\right)+\epsilon \left(\frac{k}{2}-q\right)\\ =6-2\cos \frac{k_1}{2}\cos q_1-2\cos \frac{k_2}{2}\cos q_2-2\cos \frac{k_3}{2}\cos q_3\mathrm{.}\end{array}$ (4)

From (3) and (4), it follows that

$H\left(k_1,k_2,k_3\right)=H\left(-k_1,k_2,k_3\right)=H\left(k_1,-k_2,k_3\right)=H\left(k_1,k_2,-k_3\right),$

so we can assume $k_1\mathrm{,}{k}_2\mathrm{,}{k}_3\in \left[\mathrm{0,}\pi \right]$.

The perturbation operator V is an integral operator in $L_2^o\left({\mathbb{T}}^3\right)$ with the kernel

${\left(2\pi \right)}^{-\frac{3}{2}}v\left(q-s\right)={\left(2\pi \right)}^{-\frac{3}{2}}\left(F\hat{v}\right)\left(q-s\right)\mathrm{,}$

and belongs to the class of Hilbert-Schmidt operators ${\Sigma }_2$.

In this work, we consider the operator $H\left(k\right)$ with the potential $\hat{v}$ of the form

$\hat{v}\left(n\right)=\hat{v}\left(n_1\mathrm{,}{n}_2\mathrm{,}{n}_3\right)=(\begin{array}{ll}\bar{v}\left(\left|n\right|\right)\mathrm{,}\hfill & \left|n_1\right|+\left|n_2\right|\le 1\hfill \\ \mathrm{0,}\hfill & \left|n_1\right|+\left|n_2\right|\ge 2\hfill \end{array}$ (5)

where $\left|n\right|=\left|n_1\right|+\left|n_2\right|+\left|n_3\right|$. Supporter is in the cylinder:

$D=\left\{n=\left(n_1\mathrm{,}{n}_2\mathrm{,}{n}_3\right)\in {\mathbb{Z}}^3\mathrm{<mo>\; :\; </mo>}{n}_3\in \mathbb{Z}\mathrm{,}\left|n_1\right|+\left|n_2\right|\le 1\right\}\mathrm{.}$

Since for every function $\hat{\psi }\in {\mathcal{l}}_2^{as}\left({\left({\mathbb{Z}}^3\right)}^2\right)$ the equality $\hat{\psi }\left(x\mathrm{,}x\right)=\mathrm{0,}x\in {\mathbb{Z}}^3$ holds, then the value of the potential $\hat{v}$ at the origin can be set arbitrary, since it does not affect the result, for simplicity, we assume that $\hat{v}\left(0\right)=0$.

The function $\bar{v}\mathrm{<mo>\; :\; </mo>}\mathbb{N}\to ℝ$ in (5) is decreasing in $\mathbb{N}$ i.e.,

$\bar{v}\left(1\right)>\bar{v}\left(2\right)>\cdots$ (6)

and belongs to ${\mathcal{l}}_2\left(\mathbb{N}\right)$. The kernel $v$, of the integral operator V, i.e., the Fourier transform $v\left(p\right)=\left(F\hat{v}\right)\left(p\right)$, of the potential $\hat{v}$, has the form

$\begin{array}{l}v\left(p\right)\mathrm{<mo>\; :\; </mo>}=\left(F\hat{v}\right)\left(p\right)=\frac{1}{{\left(2\pi \right)}^{3/2}}{\displaystyle \underset{n\in {\mathbb{Z}}^3}{\sum }}\hat{v}\left(n\right){\text{e}}^{i\left(n,p\right)}\\ =\frac{1}{{\left(2\pi \right)}^{3/2}}[2\bar{v}\left(1\right)\left(\cos p_1+\cos p_2+\cos p_3\right)\\ +2\bar{v}\left(2\right)\left(\cos 2p_3+2\cos p_1\cos p_2+2\cos p_1\cos p_3+2\cos p_2\cos p_3\right)\\ +2{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\bar{v}\left(n+2\right)(\cos \left(n+2\right)p_3+2\cos \left(n+1\right)p_3\left(\cos p_1+\cos p_2\right)\\ +4\cos p_1\cos p_2\cos n{p}_3)]\mathrm{.}\end{array}$ (7)

Eigenvalues of the operator $H\left(k\right)$. We note that the spectra of the operators $H_0\left(k\right)$ and V are known. The operator $H_0\left(k\right)$ does not have eigenvalues, its spectrum is continuous and coincides with the range of the function ${\epsilon }_k$ :

$\sigma \left(H_0\left(k\right)\right)=\left[m\left(k\right)\mathrm{,}M\left(k\right)\right]\mathrm{,}\text{where}m\left(k\right)=\underset{q\in {\mathbb{T}}^3}{\min }{\epsilon }_k\left(q\right)\mathrm{,}M\left(k\right)=\underset{q\in {\mathbb{T}}^3}{\max }{\epsilon }_k\left(q\right)\mathrm{.}$

The spectrum of V consists of the set $\left\{\mathrm{0,}\bar{v}\left(n\right)\mathrm{,}n\in \mathbb{N}\right\}$. Under condition (2), the operator V is a Hilbert-Schmidt operator and is hence compact. By the Weyl theorem [10], the essential spectrum of $H\left(k\right)$ coincides with the spectrum of $H_0\left(k\right)$ :

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

If $k=\stackrel{\to }{\pi }$, then the spectrum of $H\left(\stackrel{\to }{\pi }\right)=6I-V$ consists of eigenvalues of the form $6-\bar{v}\left(n\right)\mathrm{,}n\in \mathbb{N}$ and the essential spectrum is $\left\{6\right\}$. If $k_j=\pi$ (for some $j\in \left\{\mathrm{1,2,3}\right\}$ ), then there exists a potential $\hat{v}$ such that $H\left(k\right)$ has an infinite number of eigenvalues outside the continuous spectrum (see [4] [14] ).

We recall some notations and known facts. For any self-adjoint operator B acting in a Hilbert space $H$ without an essential spectrum to the right of $\mu \in ℝ$, we let $n\left(\mu \mathrm{,}B\right)$ denote the number of its eigenvalues to the right of $\mu$. We let $N\left(k\mathrm{,}z\right)$ denote the number of eigenvalues of $H\left(k\right)$ to the left of $z\le m\left(k\right)$, i.e., $N\left(k\mathrm{,}z\right)=n\left(-z\mathrm{,}-H\left(k\right)\right)$. The number $N\left(k\mathrm{,}m\left(k\right)\right)$ in fact coincides with the number of eigenvalues outside the continuous spectrum of $H\left(k\right)$. It follows from the self-adjointness of $H\left(k\right)=H_0\left(k\right)-V$ and positivity of V that

$\sigma \left(H\left(k\right)\right)\cap \left(M\left(k\right)\mathrm{,}\infty \right)=\varnothing \mathrm{,}$

and hence ${\sigma }_{disc}\left(H\left(k\right)\right)\subset \left(-\infty \mathrm{,}m\left(k\right)\right)$. Therefore we seek only eigenvalues z less than $m\left(k\right)$.

For any $k\in {\mathbb{T}}^3$ and $z<m\left(k\right)$, we define the integral operator

$G\left(k\mathrm{,}z\right)=V^{\frac{1}{2}}{r}_0\left(k\mathrm{,}z\right)V^{\frac{1}{2}}\mathrm{,}$

where $r_0\left(k\mathrm{,}z\right)$ is the resolvent of the unperturbed operator $H_0\left(k\right)$. Under condition (2), the operator V is positive, and we let $V^{\frac{1}{2}}$ denote the positive square root of the positive operator V. A solution $f$ of the Schrödinger equation

$H\left(k\right)f=zf$

and the fixed points $\phi$ of $G\left(k\mathrm{,}z\right)$ are connected by the relations

$f=r_0\left(k\mathrm{,}z\right)V^{\frac{1}{2}}\phi \text{and}\phi =V^{\frac{1}{2}}f\mathrm{.}$

The following proposition (the Birman-Schwinger principle) holds [9].

Lemma 1. The number of eigenvalues of $H\left(k\right)$ to the left of $z<m\left(k\right)$ coincides with the number of eigenvalues of $G\left(k\mathrm{,}z\right)$ greater than unity, i.e., the equality

$N\left(k\mathrm{,}z\right)=n\left(\mathrm{1,}G\left(k\mathrm{,}z\right)\right)$

holds.

Lemma 2. If for some $k\in {\mathbb{T}}^3$ the limit operator $\underset{z\to m\left(k\right)-}{\lim }G\left(k\mathrm{,}z\right)=G\left(k\mathrm{,}m\left(k\right)\right)$ exists and is compact, then the equality

$N\left(k\mathrm{,}m\left(k\right)\right)=n\left(\mathrm{1,}G\left(k\mathrm{,}m\left(k\right)\right)\right)$ (8)

holds.

Equality (8) states that the number of eigenvalues of $H\left(k\right)$, to the left of $m\left(k\right)$ is equal to the number of eigenvalues of $G\left(k\mathrm{,}m\left(k\right)\right)$ greater than unity.

3. Invariant Subspaces of $H(\; k\; )$

In this section, we study the invariant subspaces with respect to the operator $H\left(k\right)$.

Let $L_2^{-}\left(\mathbb{T}\right)=\left\{f\in L_2\left(\mathbb{T}\right)\mathrm{<mo>\; :\; </mo>}f\left(-p\right)=-f\left(p\right)\right\}$ be a subspace of the space $L_2\left(\mathbb{T}\right)$, consisting of odd functions on $\mathbb{T}=\left[-\pi \mathrm{,}\pi \right]$, and $L_2^{+}\left(\mathbb{T}\right)=\left\{f\in L_2\left(\mathbb{T}\right)\mathrm{<mo>\; :\; </mo>}f\left(-p\right)=f\left(p\right)\right\}$ be a subspace of $L_2\left(\mathbb{T}\right)$, consisting of even functions on $\mathbb{T}$. In addition, we use the notation

$L_{123}^{-}\left({\mathbb{T}}^3\right)\mathrm{<mo>\; :\; </mo>}=L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)\mathrm{,}{L}_{123}^{+}\left({\mathbb{T}}^3\right)\mathrm{<mo>\; :\; </mo>}=L_2^{+}\left(\mathbb{T}\right)\otimes L_2^{+}\left(\mathbb{T}\right)\otimes L_2^{+}\left(\mathbb{T}\right)\mathrm{.}$

Note that $L_{123}^{-}\left({\mathbb{T}}^3\right)$ is a subspace of the space $L_2^o\left({\mathbb{T}}^3\right)$. It is natural to expect the invariance of the subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$ with respect to the operator $H\left(k\right)$. It turns out that this subspace is invariant under the operator $H\left(k\right)$, i.e. the following statement holds.

Lemma 3. Let the potential $\hat{v}$ have the form (5). Then the subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$ is invariant under the action of $H\left(k\right)$ .

Proof. We prove that this subspace is invariant first with respect to $H_0\left(k\right)$, and then with respect to V. It follows from representation (4) that the function ${\epsilon }_k$ belongs to the subspace $L_{123}^{+}\left({\mathbb{T}}^3\right)$, and it follows from the inclusion $f\in L_{123}^{-}\left({\mathbb{T}}^3\right)$ that ${\epsilon }_{k}f\in L_{123}^{-}\left({\mathbb{T}}^3\right)$. This proves that $L_{123}^{-}\left({\mathbb{T}}^3\right)$ is invariant with respect to $H_0\left(k\right)$.

Simple calculations show that the function (see (7))

$\left(Vf\right)\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=\frac{1}{{\left(2\pi \right)}^{\frac{3}{2}}}{\displaystyle \underset{{\mathbb{T}}^3}{\int }}v\left(p_1-s_1\mathrm{,}{p}_2-s_2\mathrm{,}{p}_3-s_3\right)f\left(s_1\mathrm{,}{s}_2\mathrm{,}{s}_3\right)\text{d}{s}_1\text{d}{s}_2\text{d}{s}_3$

belongs to the subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$ for $f\in L_{123}^{-}\left({\mathbb{T}}^3\right)$. Hence, we prove the invariance of $L_{123}^{-}\left({\mathbb{T}}^3\right)$ with respect to V, and it follows that $L_{123}^{-}\left({\mathbb{T}}^3\right)$ is invariant with respect to $H\left(k\right)=H_0\left(k\right)-V$.

$H_{123}^{-}\left(k\right)$ denotes the restriction of $H\left(k\right)$ to the respective subspace $L_{123}^{-}\left({\mathbb{T}}^3\right)$. The action of $H_{0\left(123\right)}^{-}\left(k\right)\mathrm{<mo>\; :\; </mo>}=H_0\left(k\right)$ is unchanged, the unperturbed operator $H_0\left(k\right)$ is an operator of multiplication by the function ${\epsilon }_k$. We present the formula for $V_{123}^{-}={V|}_{L_{123}^{-}\left({\mathbb{T}}^3\right)}$ operator V acts on the element $f\in L_{123}^{-}\left({\mathbb{T}}^3\right)$ according to the formula

$\left(V_{123}^{-}f\right)\left(p\right)=\frac{1}{{\pi }^3}{\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\bar{v}\left(n+2\right){\displaystyle {\int }_{{\mathbb{T}}^3}}\sin p_1\sin p_2\sin n{p}_3\sin q_1\sin q_2\sin n{q}_{3}f\left(q\right)\text{d}q\mathrm{.}$

Note that for $k=\stackrel{\to }{\pi }$, the spectrum of $H\left(\stackrel{\to }{\pi }\right)=6I-V$ consists only of the eigenvalues $\mathrm{6,6}-\bar{v}\left(n\right)\mathrm{,}n\in \mathbb{N}$ and the essential spectrum $\left\{6\right\}$. Under condition (6) the number $z_1\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(1\right)$ is a threefold eigenvalue of $H\left(\stackrel{\to }{\pi }\right)$, with the corresponding eigenfunctions

$\sin p_1\mathrm{,}\sin p_2\mathrm{,}\sin p_3\mathrm{,}$

the number $z_2\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(2\right)$ is a sevenfold eigenvalue with the corresponding eigenfunctions

$\begin{array}{l}\sin 2p_3,\cos p_1\sin p_2,\sin p_1\cos p_2,\cos p_1\sin p_3,\\ \sin p_1\cos p_3,\cos p_2\sin p_3,\sin p_2\cos p_3,\end{array}$

for each $n\ge 3$, the number $z_n\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n\right)$ is a ninefold eigenvalue, and the corresponding eigenfunctions are

$\begin{array}{l}\sin \left(n+2\right)p_3,\sin p_1\cos \left(n+1\right)p_3,\sin p_2\cos \left(n+1\right)p_3,\\ \sin \left(n+1\right)p_3\cos p_1,\sin \left(n+1\right)p_3\cos p_2,\sin n{p}_3\cos p_1\cos p_2,\\ \sin p_2\cos p_1\cos n{p}_3,\sin p_1\cos p_2\cos n{p}_3,\sin p_1\sin p_2\sin n{p}_3.\end{array}$

The number $z_{\infty }\left(\stackrel{\to }{\pi }\right)=6$ is an eigenvalue of an infinite multiplicity, and the corresponding eigenfunctions are

${\psi }_{\left(n_1\mathrm{,}{n}_2\mathrm{,}{n}_3\right)}^{---}\left(p\right)=\sin n_1{p}_1\sin n_2{p}_2\sin n_3{p}_3\mathrm{,}{n}_3\in \mathbb{N}\mathrm{,}{n}_1+n_2\ge 3.$

All ninefold eigenvalues $z_n\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n\right)\mathrm{,}n\ge 3$ of the operator $H\left(\stackrel{\to }{\pi }\right)$ are simple eigenvalues for the operator $H_{123}^{-}\left(\stackrel{\to }{\pi }\right)$, and the number $z_{\infty }\left(\stackrel{\to }{\pi }\right)=6$ is an eigenvalue of an infinite multiplicity.

If the third coordinate $k_3$ of the total quasimomentum $k$ is equal to $\pi$, then the operator $H\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ has infinitely many invariant subspaces ${\Re }_{123}^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$.

Next, we give a description of the invariant subspace ${\Re }_{123}^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$.

The system of functions

${\left\{{\psi }_n^{-}\left(q\right)=\frac{1}{\sqrt{\pi }}\sin nq\right\}}_{n\in \mathbb{N}}$

is an orthonormal basis in the space $L_2^{-}\left(\mathbb{T}\right)$. Let us denote by $L^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$ the one-dimensional subspace spanned by the vector ${\psi }_n^{-}$. The space $L_2^{-}\left(\mathbb{T}\right)$ can be decomposed into the direct sum

$L_2^{-}\left(\mathbb{T}\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\oplus L^{-}\left(n\right)\mathrm{.}$

This decomposition produces another decomposition

$\begin{array}{c}{L}_{123}^{-}\left({\mathbb{T}}^3\right)={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\oplus \left\{L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(n\right)\right\}\\ ={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\oplus \left\{L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(n\right)\right\}={\displaystyle \underset{n=1}{\overset{\infty }{\sum }}}\oplus {\Re }_{123}^{-}\left(n\right)\mathrm{,}\end{array}$

where

${\Re }_{123}^{-}\left(n\right)\mathrm{<mo>\; :\; </mo>}=L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(n\right)\mathrm{,}{L}_{12}^{-}\left({\mathbb{T}}^2\right)=L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)\mathrm{.}$

Lemma 4. Let the potential $\hat{v}$ have the form (5). Then the subspace ${\Re }_{123}^{-}\left(n\right)$ is invariant under $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ for any $n\in \mathbb{N}$.

Proof. Let $\left(f{\psi }_n^{-}\right)\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)\mathrm{<mo>\; :\; </mo>}=f\left(p_1\mathrm{,}{p}_2\right){\psi }_n^{-}\left(p_3\right)$, where $f\in L_{12}^{-}\left({\mathbb{T}}^2\right)$, ${\psi }_n^{-}\in L^{-}\left(n\right)$ is an arbitrary element of ${\Re }_{123}^{-}\left(n\right)$. We consider the action of $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)=H_0\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)-V_{123}^{-}$ on $f{\psi }_n^{-}$ :

$\begin{array}{l}\left(H_0\mathrm{<mo\; stretchy="false">\; (\; </mo>}{k}_1\mathrm{,}{k}_2\mathrm{,}\pi \mathrm{<mo\; stretchy="false">\; )\; </mo>}f{\psi }_n^{-}\right)\left(p\right)\\ =\left[\left(6-2\cos \frac{k_1}{2}\cos p_1-2\cos \frac{k_2}{2}\cos p_2\right)f\left(p_1\mathrm{,}{p}_2\right)\right]{\psi }_n^{-}\left(p_3\right)\mathrm{,}\end{array}$ (9)

$\begin{array}{l}\left(V_{123}^{-}f{\psi }_n^{-}\right)\left(p\right)\\ =\left[\frac{\bar{v}\left(n+2\right)}{{\pi }^2}{\displaystyle {\int }_{{\mathbb{T}}^2}}\sin p_1\sin q_1\sin p_2\sin q_{2}f\left(q_1\mathrm{,}{q}_2\right)\text{d}{q}_1\text{d}{q}_2\right]{\psi }_n^{-}\left(p_3\right)\mathrm{.}\end{array}$ (10)

To obtain the last formula (10), we use the orthogonality of the system of functions ${\left\{{\psi }_n^{-}\right\}}_{n\in \mathbb{N}}$ in $L_2^{-}\left(\mathbb{T}\right)$. Relations (9) and (10) imply the equality

$\begin{array}{l}\left(H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)f{\psi }_n^{-}\right)\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)\\ =\left(H_0\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)f{\psi }_n^{-}\right)\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)-\left(V_{123}^{-}f{\psi }_n^{-}\right)\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)\\ =\left[\left(6-2\cos \frac{k_1}{2}\cos p_1-2\cos \frac{k_2}{2}\cos p_2\right)f\left(p_1\mathrm{,}{p}_2\right)\right]{\psi }_n^{-}\left(p_3\right)\\ -\left[\frac{\bar{v}\left(n+2\right)}{{\pi }^2}{\displaystyle \underset{{\mathbb{T}}^2}{\int }}\sin p_1\sin q_1\sin p_2\sin q_{2}f\left(q_1\mathrm{,}{q}_2\right)\text{d}{q}_1\text{d}{q}_2\right]{\psi }_n^{-}\left(p_3\right)\end{array}$ (11)

which completes the proof of the lemma.

We denote by $H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ restriction of the operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ in the invariant subspace ${\Re }_{123}^{-}\left(n\right)$. Formula (11) shows that the restriction $H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ to the subspace ${\Re }_{123}^{-}\left(n\right)=L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(n\right)$ has the form

$H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)=\left[2I+H_0\left(k_1\mathrm{,}{k}_2\right)-\bar{v}\left(n+2\right)V_{11}\right]\otimes I\mathrm{,}$ (12)

where I is the identity operator and $H_{123}^{\left(n\right)}\left(k\right)\mathrm{<mo>\; :\; </mo>}=2I+H_0\left(k\right)-\bar{v}\left(n+2\right)V_{11}$, $k=\left(k_1\mathrm{,}{k}_2\right)$, is a two-dimensional two-particle operator acting in $L_{12}^{-}\left({\mathbb{T}}^2\right)$ by the formula

$\begin{array}{l}\left(H_{123}^{\left(n\right)}\left(k\right)f\right)\left(p\right)\\ =\left(2+{\epsilon }_k\left(p\right)\right)f\left(p\right)-\frac{\bar{v}\left(n+2\right)}{{\pi }^2}{\displaystyle \underset{{\mathbb{T}}^2}{\int }}\sin p_1\sin p_2\sin q_1\sin q_{2}f\left(q\right)\text{d}q\mathrm{,}\end{array}$

where ${\epsilon }_k\left(p\right)=4-2\cos \frac{k_1}{2}\cos p_1-2\cos \frac{k_2}{2}\cos p_2$, and $V_{11}$ is a one-dimensional integral operator in $L_{12}^{-}\left({\mathbb{T}}^2\right)$ with the kernel

$v\left(p\mathrm{,}q\right)=\frac{1}{{\pi }^2}\sin p_1\sin p_2\sin q_1\sin q_2\mathrm{.}$

Studying the eigenvalues of $H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ by representations (12) reduces to studying the eigenvalues of

$H_{123}^{\left(n\right)}\left(k\right)=2I+H_0\left(k\right)-\bar{v}\left(n+2\right)V_{11}\mathrm{,}k=\left(k_1\mathrm{,}{k}_2\right)$

i.e. the three-dimensional problem reduces to the two-dimensional problem.

4. Eigenvalues of the Operator $H_{123}^{-}(\; k\; )$

Our main goal in this section is to study the behavior of the nondegenerate eigenvalue $z_{n+2}\left(\stackrel{\to }{\pi }\right)=6-\bar{v}\left(n+2\right)\mathrm{,}n\in \mathbb{N}$ of $H_{123}^{-}\left(\stackrel{\to }{\pi }\right)$ at small perturbations $\beta$ ( $k_1=\pi -2\beta$ or $k_2=\pi -2\beta$ ), i.e. the eigenvalues of $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ (or $H_{123}^{-}\left(\pi \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$ ) at small perturbations $\beta$. The studying of the eigenvalues of $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ is reduced to study the eigenvalues of the operator $H_{123n}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ for each fixed $n\in \mathbb{N}$. In turn, the problem of studying the eigenvalues of the operator $H_{123n}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ by virtue of (12) is reduced to study of the discrete spectrum of the operator

$H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)=2I+H_0\left(\pi -2\beta \mathrm{,}\pi \right)-\bar{v}\left(n+2\right)V_{11}\mathrm{.}$

Studying the eigenvalues of $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ and $H_{123}^{\left(n\right)}\left(\pi \mathrm{,}\pi -2\beta \right)$ reduces to studying the eigenvalues of $H_{\lambda }\left(k\right)$ acting in $L_2^{-}\left(\mathbb{T}\right)$ by the formula

$\begin{array}{l}\left(H_{\lambda }\left(k\right)f\right)\left(p\right)={\epsilon }_k\left(p\right)f\left(p\right)-\frac{\lambda }{\pi }{\displaystyle \underset{\mathbb{T}}{\int }}\sin p\sin qf\left(q\right)\text{d}q\mathrm{,}\\ {\epsilon }_k\left(p\right)=2-2\cos \frac{k}{2}\cos p\mathrm{.}\end{array}$ (13)

It is known that the essential spectrum of $H_{\lambda }\left(\pi -2\beta \right)=H_0\left(\pi -2\beta \right)-\lambda V_1\mathrm{,}\beta \in \left(\mathrm{0,}\frac{\pi }{2}\right]$ consists of a segment $\left[m\left(\beta \right)\mathrm{,}M\left(\beta \right)\right]$, where $m\left(\beta \right)=2-2\sin \beta$, $M\left(\beta \right)=2+2\sin \beta$.

Further we give some information about the eigenvalues and eigenfunctions of the operator $H_{\lambda }\left(k\right)$. Combining Theorem 6.3 in [6], Theorem 5.10 in [15] and Lemmas 1 and 2 we obtain the following statement about eigenvalues of the operator $H_{\lambda }\left(k\right)$.

Lemma 5. Let $\beta \in \left(\mathrm{0,}\frac{\pi }{2}\right]$ .

a) If $\lambda <\sin \beta$, then the operator $H_{\lambda }\left(\pi -2\beta \right)$ has no eigenvalues lying outside of the essential spectrum.

b) If $\lambda =\sin \beta$, then the left edge $m\left(\beta \right)$ of essential spectrum of the operator $H_{\lambda }\left(\pi -2\beta \right)$ is a resonance.

c) If $\lambda >\sin \beta$, then the operator $H_{\lambda }\left(\pi -2\beta \right)$ has a unique nondegenerate eigenvalue

$z_{\lambda }\left(\beta \right)=2-\lambda -\frac{1}{\lambda }{\sin }^2\beta$

which lying in the left of the essential spectrum with corresponding normalized eigenfunction

$f_{\lambda }^{-}\left(p\right)=\frac{C_{\lambda }\sin p}{2-2\sin \beta \cos p-z_{\lambda }\left(\beta \right)}\in L_2^{-}\left(\mathbb{T}\right)\mathrm{.}$ (14)

Here $C_{\lambda }$ is the normalizing multiplicity.

d) The operator $H_{\lambda }\left(\pi -2\beta \right)$ has no embedded eigenvalues in the interval $\left(m\left(\beta \right)\mathrm{,}M\left(\beta \right)\right)$.

Hilbert space $L_{12}^{-}\left({\mathbb{T}}^2\right)=L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)$ can be written as a direct sum:

$L_2^{-}\left(\mathbb{T}\right)\otimes L_2^{-}\left(\mathbb{T}\right)=L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\oplus {\left(L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\right)}^{\perp }\mathrm{.}$

The following lemma establishes a connection between the operators $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ and $H_{\lambda }\left(k\right)$.

Lemma 6. Let the potential $\hat{v}$ have the form (5). Then:

a) the subspace $L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)$ and its orthogonal complement ${\left(L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\right)}^{\perp }$ are invariant under $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$.

b) restriction of the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ to the invariant subspace ${\left(L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\right)}^{\perp }$ coinsides with the unperturbed operator $H_0\left(\pi -2\beta \mathrm{,}\pi \right)$.

c) restriction of the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ to the invariant subspace $L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)$ can be represented as a tensor product:

$H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)=\left[4I+H_0\left(\pi -2\beta \right)-\bar{v}\left(n+2\right)V_1\right]\otimes I\mathrm{.}$ (15)

Here, I is the identity operator, and $H_{\lambda \left(n\right)}\left(\pi -2\beta \right)\mathrm{<mo>\; :\; </mo>}=H_0\left(\pi -2\beta \right)-\lambda \left(n\right)V_1$, $\lambda \left(n\right)=\bar{v}\left(n+2\right)$ is a one-dimensional two-particle operator acting in $L_2^{-}\left(\mathbb{T}\right)$ by the formula (13).

This lemma is proved in the same way as the Lemma 4. In particular, part b) of the lemma implies that the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ has no eigenfunctions in ${\left(L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\right)}^{\perp }$. Thus, studying the eigenvalues of the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ is reduced to studying eigenvalues of the operator $H_{\lambda \left(n\right)}\left(\pi -2\beta \right)=H_0\left(\pi -2\beta \right)-\lambda \left(n\right)V_1$.

From Lemmas 5 - 6 and tensor product (15) implies the following statement regarding operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$.

Theorem 1. Let $\beta \in \left(\mathrm{0,}\frac{\pi }{2}\right]$ and $n\in \mathbb{N}$ .

a) If $\bar{v}\left(n+2\right)<\sin \beta$, then the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ has no eigenvalues lying outside of the essential spectrum.

b) If $\bar{v}\left(n+2\right)=\sin \beta$, then the left edge $m\left(\beta \right)$ of essential spectrum of the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ is a resonance.

c) If $\bar{v}\left(n+2\right)>\sin \beta$, then the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ has a unique nondegenerate eigenvalue

$z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)=4+z_{\lambda \left(n\right)}\left(\beta \right)=6-\bar{v}\left(n+2\right)-\frac{1}{\bar{v}\left(n+2\right)}{\sin }^2\beta \mathrm{,}$ (16)

which lies in the left of the essential spectrum and with the corresponding normalized eigenfunction

$f_{\lambda \left(n\right)}^{--}\left(p_1\mathrm{,}{p}_2\right)=f_{\lambda \left(n\right)}^{-}\left(p_1\right)\frac{\sin p_2}{\sqrt{\pi }}=f_{\lambda \left(n\right)}^{-}\left(p_1\right){\psi }_1^{-}\left(p_2\right)\in L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\mathrm{,}$

where $f_{\lambda \left(n\right)}^{-}$ is the normalized eigenfunction of the operator $H_{\lambda \left(n\right)}\left(\pi -2\beta \right)$ corresponding to the eigenvalue $z_{\lambda \left(n\right)}\left(\beta \right)$, the operator $H_{\lambda \left(n\right)}\left(k\right)$ is defined by the formula (13).

d) The operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ has no embedded eigenvalues in the interval $\left(m\left(\beta \right)\mathrm{,}M\left(\beta \right)\right)$.

Similar statement is true for the operator $H_{123}^{\left(n\right)}\left(\pi \mathrm{,}\pi -2\beta \right)$. The eigenvalues of the operators $H_{123}^{\left(n\right)}\left(\pi \mathrm{,}\pi -2\beta \right)$ and $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ are same, but eigenfunctions differ with variable replacement $p_1$ and $p_2$. In other words, the operators $H_{123}^{\left(n\right)}\left(k_1\mathrm{,}{k}_2\right)$ and $H_{123}^{\left(n\right)}\left(k_2\mathrm{,}{k}_1\right)$ are unitary equivalent. Therefore, the operators $H_{123n}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ and $H_{123n}^{-}\left(k_2\mathrm{,}{k}_1\mathrm{,}\pi \right)$ are unitary equivalent too.

Similar statement can relatively be formulated for the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$. For this purpose, we introduce the following notation. Through

${\Delta }_n\left(\beta \mathrm{,}z\right)=1-\frac{\bar{v}\left(n+2\right)}{{\pi }^2}{\displaystyle {\int }_{{\mathbb{T}}^2}}\frac{{\sin }^2{p}_1{\sin }^2{p}_2\text{d}{p}_1\text{d}{p}_2}{2+2\left(2-\sin \beta \cos p_1-\sin \beta \cos p_2\right)-z}$

we denote the Fredholm determinant of the operator $I-\bar{v}\left(n+2\right)V_{11}{r}_0\left(\beta \mathrm{,}z\right)$, where $r_0\left(\beta \mathrm{,}z\right)$ is the resolvent of the operator $2I+H_0\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$, and $V_{11}$ is an integral operator with the kernel

$v\left(p\mathrm{,}q\right)=\frac{1}{{\pi }^2}\sin p_1\sin p_2\sin q_1\sin q_2\mathrm{.}$

Through $C_{11}^{--}$ denote the value of the following integral:

$C_{11}^{--}=\frac{1}{{\pi }^2}{\displaystyle {\int }_{{\mathbb{T}}^2}}\frac{{\sin }^2{p}_1{\sin }^2{p}_2\text{d}{p}_1\text{d}{p}_2}{2\left(2-\cos p_1-\cos p_2\right)}={\displaystyle {\int }_{{\mathbb{T}}^2}}\frac{{\left|{\psi }_1^{-}\left(p_1\right)\right|}^2{\left|{\psi }_1^{-}\left(p_2\right)\right|}^2\text{d}{p}_1\text{d}{p}_2}{2\epsilon \left(p\right)}\mathrm{.}$

Simple calculations reveal the following approximate value $C_{11}^{--}\approx 0.302347$.

Theorem 2. Let $\beta \in \left(\mathrm{0,}\frac{\pi }{2}\right]$ , $n\in \mathbb{N}$.

a) If $\bar{v}\left(n+2\right)<\frac{\sin \beta }{C_{11}^{--}}$, then the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ has no eigenvalues lying outside of the essential spectrum.

b) If $\bar{v}\left(n+2\right)=\frac{\sin \beta }{C_{11}^{--}}$, then the left edge $m\left(\beta \right)=6-4\sin \beta$ of the spectrum of the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ is an eigenvalue.

c) If $\bar{v}\left(n+2\right)>\frac{\sin \beta }{C_{11}^{--}}$, then the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ has a unique nondegenerate eigenvalue $z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ below the essential spectrum.

d) The operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ has no embedded eigenvalues in the interval $\left(m\left(\beta \right)\mathrm{,}M\left(\beta \right)\right)$.

This theorem is proved in similar way as Lemma 5. There are some differences:

1) In the Theorem 2, the eigenvalue $z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ was calculated with the accuracy of ${\beta }^2$ :

$z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)=6-\bar{v}\left(n+2\right)-\frac{2}{\bar{v}\left(n+2\right)}{\sin }^2\beta +O(\; \beta \; 4\; )$

and corresponding normalized eigenfunction has the form

$\begin{array}{l}{f}_{123}^{\left(n\right)}\left(p_1\mathrm{,}{p}_2\right)\\ =\frac{C_n\left(\beta \right)\sin p_1\sin p_2}{6-2\sin \beta \cos p_1-2\sin \beta \cos p_2-z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)}\in L_{12}^{-}\left({\mathbb{T}}^2\right)\mathrm{,}\end{array}$ (17)

where $C_n\left(\beta \right)$ is the normalizing multiplicity.

2) Left edge $m\left(\beta \right)=6-2\sin \beta$ of the essential spectrum is a resonance for the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$, but for the operator $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ the left edge $m\left(\beta \right)=6-4\sin \beta$ of the essential spectrum is the eigenvalue, i.e. the equation $H_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)f=m\left(\beta \right)f$ has a non-trivial solution

$f\left(p_1\mathrm{,}{p}_2\right)=\frac{C\sin p_1\sin p_2}{2-\cos p_1-\cos p_2}$

and it belongs to $L_{12}^{-}\left({\mathbb{T}}^2\right)$.

5. Conclusions

1) We have shown that the operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}\pi \right)$ has infinitely many invariant subspaces ${\Re }_{123}^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$. It has been proved that if condition $\bar{v}\left(n+2\right)>\sin \beta$ holds then the operator $H_{123n}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ has a unique simple eigenvalue $z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)$ of the form (16), otherwise, the operator has no eigenvalues outside of the essential spectrum. A similar statement holds for the operator $H_{123n}^{-}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$.

2) Without loss of generality it can be assumed that $\bar{v}\left(3\right)\le 1$. Since, if $\bar{v}\left(3\right)>1$ then it follows from $\underset{n\to \infty }{lim}\bar{v}\left(n\right)=0$ that there exists a number $m\in \mathbb{N}$ such that $\bar{v}\left(m+2\right)\le 1$ and monotonicity of $\bar{v}$ implies that $\bar{v}\left(n\right)>1$ for $n=3,4,\cdots ,m+1$, and in this case, the eigenvalues $z_{123}^{\left(n\right)}\left(\pi -2\beta \mathrm{,}\pi \right)\mathrm{,}n=\mathrm{1,2,}\cdots \mathrm{,}m-1$ of $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ exist for all $\beta \in \left[\mathrm{0,}\pi /2\right]$.

For a fixed $\beta \in \left(\mathrm{0,}\pi /2\right]$ there exists $m\in \mathbb{N}$ such that $\sin \beta \in \left(\bar{v}\left(m+3\right)\mathrm{,}\bar{v}\left(m+2\right)\right)$ and the operator $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ has m nondegenerate eigenvalues outside of the essential spectrum (see Theorem 1):

$z_{123}^{\left(1\right)}\left(\pi -2\beta ,\pi ,\pi \right):=z_{123}^{\left(1\right)}\left(\pi -2\beta ,\pi \right)=6-\bar{v}\left(3\right)-\frac{1}{\bar{v}\left(3\right)}{\sin }^2\beta ,$

$z_{123}^{\left(2\right)}\left(\pi -2\beta ,\pi ,\pi \right):=z_{123}^{\left(2\right)}\left(\pi -2\beta ,\pi \right)=6-\bar{v}\left(4\right)-\frac{1}{\bar{v}\left(4\right)}{\sin }^2\beta ,$

$⋮$

$z_{123}^{\left(m\right)}\left(\pi -2\beta ,\pi ,\pi \right):=z_{123}^{\left(m\right)}\left(\pi -2\beta ,\pi \right)=6-\bar{v}\left(m+2\right)-\frac{1}{\bar{v}\left(m+2\right)}{\sin }^2\beta .$

The corresponding normalized eigenfunctions are of the forms:

$f_{123\lambda \left(1\right)}^{---}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{\lambda \left(1\right)}^{-}\left(p_1\right){\psi }_1^{-}\left(p_2\right){\psi }_1^{-}\left(p_3\right)\in L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\otimes L^{-}\left(1\right)\mathrm{,}$

$f_{123\lambda \left(2\right)}^{---}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{\lambda \left(2\right)}^{-}\left(p_1\right){\psi }_1^{-}\left(p_2\right){\psi }_2^{-}\left(p_3\right)\in L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\otimes L^{-}\left(2\right)\mathrm{,}$

$⋮$

$f_{123\lambda \left(m\right)}^{---}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{\lambda \left(m\right)}^{-}\left(p_1\right){\psi }_1^{-}\left(p_2\right){\psi }_m^{-}\left(p_3\right)\in L_2^{-}\left(\mathbb{T}\right)\otimes L^{-}\left(1\right)\otimes L^{-}\left(m\right)\mathrm{,}$

where, $f_{\lambda \left(m\right)}^{-}$ is the normalized eigenfunction of the operator $H_{\lambda \left(m\right)}\left(\pi -2\beta \right)$ corresponding to the eigenvalue $z_{\lambda \left(m\right)}\left(\beta \right)$ and the operator $H_{\lambda \left(m\right)}\left(k\right)$ is defined by the formula (13), $\lambda \left(m\right)=\bar{v}\left(m+2\right)$.

The eigenvalues of the operators $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ and $H_{123}^{-}\left(\pi \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$ are same but eigenfunctions differ with variable replacement $p_1$ and $p_2$. In other words, the operators $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ and $H_{123}^{-}\left(\pi \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$ are unitary equivalent.

In the case $\sin \beta =\bar{v}\left(m+2\right)$, the left edge $m\left(\beta \right)=6-2\sin \beta$ of the essential spectrum is a resonance of the operator $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi \mathrm{,}\pi \right)$ (see Theorem 1).

3) Let for some $m\in \mathbb{N}$ the relation $\sin \beta \in \left(\bar{v}\left(m+3\right)C_{11}^{--}\mathrm{,}\bar{v}\left(m+2\right)C_{11}^{--}\right)$ hold then the operator $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$ has m nondegenerate eigenvalues outside the essential spectrum (see Theorem 2) and for small $\beta$ :

$\begin{array}{l}{z}_{123}^{\left(1\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)\mathrm{<mo>\; :\; </mo>}=z_{123}^{\left(1\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)\\ =6-\bar{v}\left(3\right)-\frac{2}{\bar{v}\left(3\right)}{\sin }^2\beta +O\left({\beta }^4\right)\mathrm{,}\end{array}$

$\begin{array}{l}{z}_{123}^{\left(2\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)\mathrm{<mo>\; :\; </mo>}=z_{123}^{\left(2\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)\\ =6-\bar{v}\left(4\right)-\frac{2}{\bar{v}\left(4\right)}{\sin }^2\beta +O\left({\beta }^4\right)\mathrm{,}\end{array}$

$⋮$

$\begin{array}{l}{z}_{123}^{\left(m\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)\mathrm{<mo>\; :\; </mo>}=z_{123}^{\left(m\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)\\ =6-\bar{v}\left(m+2\right)-\frac{2}{\bar{v}\left(m+2\right)}{\sin }^2\beta +O\left({\beta }^4\right)\mathrm{.}\end{array}$

The corresponding normalized eigenfunctions are of the forms:

$f_{123}^{\left(1\right)-}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{123}^{\left(1\right)}\left(p_1\mathrm{,}{p}_2\right){\psi }_1^{-}\left(p_3\right)\in L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(1\right)\mathrm{,}$

$f_{123}^{\left(2\right)-}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{123}^{\left(2\right)}\left(p_1\mathrm{,}{p}_2\right){\psi }_2^{-}\left(p_3\right)\in L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(2\right)\mathrm{,}$

$⋮$

$f_{123}^{\left(m\right)-}\left(p_1\mathrm{,}{p}_2\mathrm{,}{p}_3\right)=f_{123}^{\left(m\right)}\left(p_1\mathrm{,}{p}_2\right){\psi }_m^{-}\left(p_3\right)\in L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(m\right)\mathrm{,}$

where, $f_{123}^{\left(m\right)}$ is the normalized eigenfunction of the operator $H_{123}^{\left(m\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ corresponding to the eigenvalue $z_{123}^{\left(m\right)}\left(\pi -2\beta \mathrm{,}\pi -2\beta \right)$ defined by the formula (17).

In the case $\sin \beta =\bar{v}\left(m+2\right)C_{11}^{--}$, the left edge $m\left(\beta \right)=6-4\sin \beta$ of the essential spectrum is the eigenvalue of $H_{123}^{-}\left(\pi -2\beta \mathrm{,}\pi -2\beta \mathrm{,}\pi \right)$ (see Theorem 2) with the corresponding eigenfunction

$f\left(p\right)=\frac{C\sin p_1\sin p_2}{2-\cos p_1-\cos p_2}\cdot \sin m{p}_3\in L_{12}^{-}\left({\mathbb{T}}^2\right)\otimes L^{-}\left(m\right)\mathrm{.}$

Remark 1. If the potential $\hat{v}$ is even in all arguments $p_1\mathrm{,}{p}_2\mathrm{,}{p}_3$ and the condition $\hat{v}\in {\mathcal{l}}_2\left({\mathbb{Z}}^3\right)$ holds, then the statements of Lemmas 3 - 4 remain valid.

Remark 2. If $k_3\ne \pi$, then the subspaces ${\Re }_{123}^{-}\left(n\right)\mathrm{,}n\in \mathbb{N}$ are not invariant under the operator $H_{123}^{-}\left(k_1\mathrm{,}{k}_2\mathrm{,}{k}_3\right)$.

Acknowledgements

This work was supported by the Grant OT-F4-66 of Fundamental Science Foundation of Uzbekistan.