# The number and location of eigenvalues for the two-particle Schrödinger
operators on lattices

Saidakhmat N. Lakaev, Mukhayyo O. Akhmadova Samarkand State University,
140104, Samarkand, Uzbekistan<EMAIL_ADDRESS>Samarkand State University,
140104, Samarkand, Uzbekistan<EMAIL_ADDRESS>
###### Abstract.

We study the Schrödinger operators $H_{\gamma\lambda\mu}(K)$, $K\in\mathbb{T}$
being a fixed (quasi)momentum of the particles pair, associated with a system
of two identical bosons on the one-dimensional lattice $\mathbb{Z}$, where the
real quantities $\gamma$, $\lambda$ and $\mu$ describe the interactions
between pairs of particles on one site, two nearest neighboring sites and next
two neighboring sites, respectively. We found a partition of the three-
dimensional space $(\gamma,\lambda,\mu)$ of interaction parameters into
connected components and the exact number of eigenvalues of this operator that
lie below and above the essential spectrum, in each component. Moreover, we
show that for any $K\in\mathbb{T}^{d}$ the number of eigenvalues of
$H_{\gamma\lambda\mu}(K)$ is not less than the corresponding number of
eigenvalues of $H_{\gamma\lambda\mu}(0)$.

###### Key words and phrases:

Two-particle system, discrete Schrödinger operator, essential spectrum, bound
states, Fredholm determinant

##  1\. Introduction

Lattice models of physical systems are one of the widely used mathematical
models in mathematical physics. Few-body Hamiltonians [29], among such models
may be viewed as the simplest version of the corresponding Bose-Hubbard model
involving a finite number of particles of a certain type. The few-body lattice
Hamiltonians have been intensively studied over the past several decades [2,
3, 4, 5, 10, 11, 17, 19, 20, 21, 22, 23, 27, 25, 30].

Another reason for studying the discrete Hamiltonians is that they can provide
a natural approximation for their continuous counterparts [9], which allows
studying few-body systems. It is well known that the Efimov effect [8] was
originally attributed to the three-body systems moving in the three-
dimensional continuous space $\mathbb{R}^{3}$. A rigorous mathematical proof
of the Efimov effect has been given in [32, 35, 36, 39]. The celebrated Efimov
phenomenon to take place also in the lattice three-particle systems [2, 4, 7,
19]. Discrete Schrödinger operators also represent a simple model for
description of few-body systems formed by particles traveling through periodic
structures, such as ultracold atoms injected into optical crystals created by
the interference of counter-propagating laser beams [6, 38]. The study of
ultracold few-atom systems in optical lattices have became popular in the last
years due to availability of controllable parameters, such as temperature,
particle masses, interaction potentials etc. (see e.g., [6, 13, 14, 28] and
references therein). These possibilities give an opportunity to experimentally
observe stable repulsively bound pairs of ultracold atoms ([31, 38], which is
not the case for the traditional condensed matter systems, where stable
composite objects are formed by means attractive forces. Lattice Hamiltonians
are of particular interest in fusion physics too. For example, in [30], a one-
dimensional one-particle lattice Hamiltonian has been successfully employed to
show that certain class of molecules in lattice structures may enhance nuclear
fusion probability.

In the continuous case, the center-of-mass motion can be separated, which is
not the case for lattice few-body problems. However, for the lattice
Hamiltonian $\mathrm{H}$ acting in the functional Hilbert space
$\mathbb{T}^{n\cdot d}$ we have a von Neumann direct integral decomposition

(1.1) $\mathrm{H}\simeq\int\limits_{K\in\mathbb{T}^{d}}^{\oplus}H(K)\,dK,$

where $\mathbb{T}^{d}$ is the $d$-dimensional torus. The so called fiber
Hamiltonians $H(K)$ acting on the Hilbert space $\mathbb{T}^{(n-1)d}$
nontrivially depends on the quasimomentum $K\in\mathbb{T}^{d}$ (see e.g., [2,
3, 33]). This decomposition allows us to reduce the problem to studying the
operators $H(K)$.

In this work, we study the spectral properties of the fiber Hamiltonians
$H(K),\,K\in\mathbb{T}$ acting in the Hilbert space $L^{2,e}(\mathbb{T})$ as

(1.2) $H_{\gamma\lambda\mu}(K):=H_{0}(K)+V_{\gamma\lambda\mu},$

where $H_{0}(K)$ is the fiber kinetic-energy operator,

$\bigl{(}H_{0}(K)f\bigr{)}(p)={\mathcal{E}}_{K}(p)f(p),$

with

(1.3) ${\mathcal{E}}_{K}(p):=2(1-\cos\tfrac{K}{2}\,\cos p)$

and $V_{\gamma\lambda\mu}$ is the combined interaction potential.The
parameters $\gamma$, $\lambda$ and $\mu$, called coupling constants, describe
interactions between the particles which are located on one site, on the
nearest neighboring sites of the lattice and in the next nearest neighboring
sites, respectively.

Within this new model, we find both the exact number of eigenvalues and their
locations of the operator $H_{\gamma\lambda\mu}(0)$. We describe the
mechanisms of emission and absorption of the eigenvalues of
$H_{\gamma\lambda\mu}(0)$ at the thresholds of its essential (continuous)
spectrum depending on the interaction parameters. Furthermore, we establish
sharp lower bounds for the number of isolated eigenvalues
$H_{\gamma\lambda\mu}(K)$ depending on the quasi-momentum $K\in\mathbb{T}$,
which lie both below the essential spectrum and above that.

For this, we apply the results obtained for the operator
$H_{\gamma\lambda\mu}(0)$ and the nontrivial dependence of the dispersion
relation ${\mathcal{E}}_{K}$ on the (quasi)momentum $K\in\mathbb{T}^{q}$.

We recall that the two-particle Schrödinger operator
${H}_{\mu}(K)=-\Delta+\mu{V},\,\mu>0$ on the lattice $\mathbb{Z}^{d}$
associated to a system of two bosons with zero-range repulsive interactions
$\mu>0$ has been considered as a theoretical basis for explanation of the
experimental results obtained in [31, 38].

Note that the continuous counterpart of the two-particle Schrödinger operators
on lattices, which associated with a system of two quantum-mechanical
particles on $\mathbb{R}^{d}$ interacting via short-range potential $v(x)$ has
isolated eigenvalues lying only below the essential spectrum fulfilling the
semi-axis $[0,+\infty)$ and hence this model is well adapted to describe
systems of two-particles with attractive interactions.

To study the eigenvalues of the discrete Schrödinger operator
$H_{\gamma\lambda\mu}(K)$, we apply analytic function theory, namely, we
investigate the corresponding Fredholm determinant
$\Delta_{\gamma\lambda\mu}(K,z)$, as there is a one-to-one mapping between the
sets of eigenvalues of the operator $H_{\gamma\lambda\mu}(K)$ and the zeros of
$\Delta_{\gamma\lambda\mu}(K,z)$ (see [23]). Correspondingly, the change in
the number of zeros of Fredholm determinant $\Delta_{\gamma\lambda\mu}(0,z)$
results in the change of the number of isolated eigenvalues of Schrödinger
operator $H_{\gamma\lambda\mu}(0)$.

Our main finding is that the number of zeros of the determinant
$\Delta_{\gamma\lambda\mu}(0,z)$ located below (resp. above) the essential
spectrum changes if and only if the principal term $C^{-}(\gamma,\lambda,\mu)$
(resp. $C^{+}(\gamma,\lambda,\mu)$) of the asymptotics of the Fredholm
determinant $\Delta_{\gamma\lambda\mu}(0,z)$ vanishes as $z$ approaches the
lower (resp. upper) threshold of the essential spectrum (see Lemma 4.3).

Using this property, we establish a partition of the three-dimensional
$(\gamma,\lambda,\mu)$-space into four disjoint connected components by means
of curves $C^{-}(\gamma,\lambda,\mu)=0$ or $C^{+}(\gamma,\lambda,\mu)=0$. This
allows us to prove that the number of zeros of the Fredholm determinant is
constant in each connected component.

In [23, 25], it was studied the Schrödinger operators on the lattice
$\mathbb{Z}^{d},\,d=1,2$, associated to a system of two bosons with the zero-
range on one site interaction ($\lambda\in\mathbb{R}$) and interactions on the
nearest neighboring sites ($\mu\in\mathbb{R}$) of the $d$\- dimensional
lattice $\mathbb{Z}^{d}$. We emphasize that, our results is an extensions of
[23, 25]. The authors of [23] consider the Schrödinger operators
$H_{\lambda\mu}(K)$ on two-dimensional lattice $\mathbb{Z}^{2}$. The operator
$H_{\lambda\mu}(K)$ can have one or two eigenvalues, lying as below the
essential spectrum, as well as above it. The connected components, which split
the two-dimensional plane $\mathbb{R}^{2}$ of interaction parameters are
described by means of a second order elementary curves (hyperbolas). Similar
results for the number of eigenvalues of one-particle Schrödinger operators in
$\mathbb{Z}^{d},\,d\geq 1$ have been obtained, for instance in [22] with
attractive interactions and $d=3$ and also with attractive and repulsive
interaction cases in [11] for all $d\geq 1$ considering only negative
eigenvalues.

In the present work, the connected components are described by implicit
functions (third-order polynomials) of three variables defined on
$\mathbb{R}^{3}$. This is a more challenging case as the connected components
are described by the third-order surfaces.

The discrete two-particle Schrödinger operator ${H}_{\mu}(k)$ associated to a
system of one and two quantum-mechanical particles on $\mathbb{Z}^{d}$
interacting via short-range potentials have been studied in recent years [3,
5, 18, 20, 30].

Note that some results such as the existence of eigenvalues and their
finiteness can be received for a large class of Schrödinger type operators
(see e.g., [15, 16, 18, 24]). However, our results show that the study of a
qualitative change in the number of eigenvalues of $H_{\gamma\mu\lambda}(K),$
even for $K=0,$ is very delicate: There is ball, with arbitrarily small
radius, in the three-dimensional $(\gamma,\lambda,\mu)$-space in which the
number of eigenvalues has a jump (see Theorem 3.6).

The paper is organized as follows: The section 1 is introduction. In Section
2, we introduce the two-particle Hamiltonian $\mathbb{H}_{\gamma\lambda\mu}$
of a system of two bosons in the position and momentum space representations
and also the Schrödinger operator $H_{\gamma\lambda\mu}(K)$ associated to the
Hamiltonian $\mathbb{H}_{\gamma\lambda\mu}$. The main results of the paper are
stated in Section 3 and their proofs are contained in Section 4.

##  2\. The two-particle Hamiltonian

###  2.1. The position-space representation

Let $\mathbb{Z}$ be the one-dimensional lattice and
$\mathbb{Z}\times\mathbb{Z}$ be cartesian square of $\mathbb{Z}$. Let
$\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})$ be the Hilbert space of square-
summable symmetric functions on $\mathbb{Z}\times\mathbb{Z}$.

The free Hamiltonian $\widehat{\mathbb{H}}_{0}$ of a system of two identical
particles (bosons), in the position space representation, is usually
associated with the following self-adjoint (bounded) Toeplitz-type operator on
the Hilbert space $\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})$:

$(\widehat{\mathbb{H}}_{0}\widehat{f})(x_{1},x_{2})=\sum_{s_{1}\in\mathbb{Z}}\widehat{\varepsilon}(x_{1}-s_{1})\widehat{f}(s_{1},x_{2})+\sum_{s_{2}\in\mathbb{Z}}\widehat{\varepsilon}(x_{2}-s_{2})\widehat{f}(x_{1},s_{2}),\,\,\widehat{f}\in\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})$

where

(2.1) $\widehat{\varepsilon}(s)=\begin{cases}2&\text{if $|s|=0,$}\\\
-\frac{1}{2}&\text{if $|s|=1,$}\\\ 0&\text{if $|s|>1.$}\end{cases}$

The interaction operator $\widehat{\mathbb{V}}_{\gamma\lambda\mu}$, in the
position space representation, is the multiplication operator by the function
$\widehat{v}\in\ell^{1}(\mathbb{Z}),i.e.,$

$\widehat{\mathbb{V}}_{\gamma\lambda\mu}\widehat{f}(x,y)=\widehat{v}_{\gamma\lambda\mu}(x-y)\widehat{f}(x,y),\,\,\widehat{f}\in\ell^{2,s}(\mathbb{Z}\times\mathbb{Z}),$

where

(2.2) $\widehat{v}_{\gamma\lambda\mu}(s)=\begin{cases}2\gamma&\text{if
$|s|=0,$}\\\ \lambda&\text{if $|s|=1,$}\\\ \mu&\text{if $|s|=2,$}\\\
0&\text{if $|s|>2.$}\end{cases}$

The total Hamiltonian $\widehat{\mathbb{H}}_{\gamma\lambda\mu}$ of a system of
two identical particles is described as a bounded self-adjoint operator on
$\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})$:

$\widehat{\mathbb{H}}_{\gamma\lambda\mu}=\widehat{\mathbb{H}}_{0}+\widehat{\mathbb{V}}_{\gamma\lambda\mu},\,\,\gamma,\lambda,\mu\in\mathbb{R}.$

###  2.2. The two-particle Hamiltonian: the (quasi)momentum-space
representation

Let $\mathbb{T}=\allowbreak\mathbb{R}/2\pi\mathbb{Z}\equiv(-\pi,\pi]$ be the
one-dimensional torus, the Pontryagin dual group of $\mathbb{Z}$. Let
$L^{2}(\mathbb{T})$ be the Hilbert space of square-integrable functions on
$\mathbb{T}$ and let

${\mathcal{F}}:\ell^{2}(\mathbb{Z})\rightarrow
L^{2}(\mathbb{T}),\,\,{\mathcal{F}}\widehat{f}(p)=\frac{1}{2\pi}\sum_{x\in\mathbb{Z}}\widehat{f}(x)e^{ipx}.$

is the standard Fourier transform with the inverse

${\mathcal{F}}^{*}:L^{2}(\mathbb{T})\rightarrow\ell^{2}(\mathbb{Z}),\,\,{\mathcal{F}}^{*}f(p)=\frac{1}{2\pi}\int_{\mathbb{T}}f(x)e^{ipx}dx.$

The free Hamiltonian
$\mathbb{H}_{0}=({\mathcal{F}}\otimes{\mathcal{F}})\widehat{\mathbb{H}}_{0}({\mathcal{F}}^{*}\otimes{\mathcal{F}}^{*})$
of a system of two identical particles,in the momentum space representation,
is the multiplication operator by the function $\varepsilon(p)+\varepsilon(q)$
in the Hilbert space $L^{2,s}(\mathbb{T}\times\mathbb{T})$ of symmetric
functions on the cartesian square $\mathbb{T}\times\mathbb{T}$ of the torus
$\mathbb{T}$:

$\mathbb{H}_{0}f(p,q)=[\varepsilon(p)+\varepsilon(q)]f(p,q),$

where the continuous function (dispersion relation) $\varepsilon$ is given by

$\varepsilon(p)=[{\mathcal{F}}\widehat{\varepsilon}](p)=1-\cos p,\quad
p\in\mathbb{T}.$

The interaction operator
$\mathbb{V}_{\gamma\lambda\mu}=({\mathcal{F}}\otimes{\mathcal{F}})\widehat{\mathbb{V}}_{\gamma\lambda\mu}({\mathcal{F}}^{*}\otimes{\mathcal{F}}^{*})$
is the integral operator of convolution type acting in
$L^{2,s}(\mathbb{T}\times\mathbb{T})$ as

$(\mathbb{V}_{\gamma\lambda\mu}f)(p,q)=\frac{1}{2\pi}\int_{\mathbb{T}}v_{\gamma\lambda\mu}(p-u)f(u,p+q-u)\mathrm{d}u,$

where the kernel function $v_{\gamma\lambda\mu}(\cdot)$ is given by

$v_{\gamma\lambda\mu}(p)=[{\mathcal{F}}\widehat{v}_{\gamma\lambda\mu}](p)=\frac{1}{2\pi}\sum_{x\in\mathbb{Z}}\widehat{v}_{\gamma\lambda\mu}(x)e^{ipx}=\frac{1}{\pi}(\gamma+\lambda\cos
p+\mu\cos 2p),\quad p\in\mathbb{T}.$

The total two-particle Hamiltonian $\mathbb{H}_{\gamma\lambda\mu}$ of a system
of two identical quantum-mechanical patricles interacting via a finite range
attractive potentials $\widehat{v}_{\gamma\lambda\mu}$, in the momentum space
representation, is the bounded self-adjoint operator acting in
$L^{2,s}(\mathbb{T}\times\mathbb{T})$ as

$\mathbb{H}_{\gamma\lambda\mu}:=({\mathcal{F}}\otimes{\mathcal{F}})\widehat{\mathbb{H}}_{\gamma\lambda\mu}({\mathcal{F}}\otimes{\mathcal{F}})^{*}=\mathbb{H}_{0}+\mathbb{V}_{\gamma\lambda\mu}.$

###  2.3. The Floquet-Bloch decomposition of $\mathbb{H}_{\gamma\lambda\mu}$
and discrete Schrödinger operators

Since the operator $\widehat{H}_{\gamma\lambda\mu}$ commutes with the shift
operators on the lattice $\mathbb{Z}\times\mathbb{Z}$, we can decompose the
space $L^{2,s}(\mathbb{T}\times\mathbb{T})$ and Hamiltonian
$\mathbb{H}_{\gamma\lambda\mu}$ into the von Neumann direct integrals as

(2.3)
$L^{2,s}(\mathbb{T}\times\mathbb{T})\simeq\int\limits_{K\in\mathbb{T}}^{\oplus}L^{2,e}(\mathbb{T})\,\mathrm{d}K\quad\text{and}\quad\mathbb{H}_{\gamma\lambda\mu}\simeq\int\limits_{K\in\mathbb{T}}^{\oplus}H_{\gamma\lambda\mu}(K)\,\mathrm{d}K,$

respectively, where $L^{2,e}(\mathbb{T})$ is the Hilbert space of square-
integrable even functions on $\mathbb{T}$ (see, e.g., [3]).

The fiber operator $H_{\gamma\lambda\mu}(K),$ $K\in\mathbb{T}$ is a self-
adjoint operator defined in $L^{2,e}(\mathbb{T})$ as

$H_{\gamma\lambda\mu}(K):=H_{0}(K)+V_{\gamma\lambda\mu},$

where the unperturbed operator $H_{0}(K)$ is the multiplication operator by
the function

${\mathcal{E}}_{K}(p):=2(1-\cos\tfrac{K}{2}\,\cos p)$

and the perturbation $V_{\gamma\lambda\mu}$ is defined as

$V_{\gamma\lambda\mu}f(p)=\frac{1}{\pi}\int_{\mathbb{T}}(\gamma+\lambda\cos
p\cos q+\mu\cos 2p\cos 2q)f(q)\mathrm{d}q.$

In some literature, the parameter $K$ is called _quasi-momentum_ and the fiber
$H_{\gamma\lambda\mu}(K)$ is called _discrete Schrödinger operator_ associated
to the two-particle Hamiltonian $\mathbb{H}_{\gamma\lambda\mu}$.

Using the Fourier transform, $\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})$ and
$\mathbb{H}_{\gamma\lambda\mu}$ can also be rewritten as decompositions

$\ell^{2,s}(\mathbb{Z}\times\mathbb{Z})\simeq\int\limits_{K\in\mathbb{T}}^{\oplus}\ell^{2,e}(\mathbb{Z})\,\mathrm{d}K\quad\text{and}\quad\mathbb{H}_{\gamma\lambda\mu}\simeq\int\limits_{K\in\mathbb{T}}^{\oplus}H_{\gamma\lambda\mu}(K)\,\mathrm{d}K,$

respectively, where $\ell^{2,e}(\mathbb{Z})$ is the Hilbert space of square-
summable even functions on $\mathbb{Z}$ and

(2.4)
$\widehat{H}_{\gamma\lambda\mu}(K):={\mathcal{F}}^{*}H_{\gamma\lambda\mu}(K){\mathcal{F}}=\widehat{H}_{0}(K)+\widehat{V}_{\gamma\lambda\mu},$

and

$\widehat{H}_{0}(K)\widehat{f}(x)\ =\
\sum_{s\in\mathbb{Z}}\widehat{\mathcal{E}}_{K}(x-s)f(s),\quad\widehat{f}\in\ell^{2,e}(\mathbb{Z}),$

with

$\widehat{\mathcal{E}}_{K}(x)=2\widehat{\varepsilon}(x)\cos\frac{Kx}{2}$

and the operator $\widehat{V}_{\gamma\lambda\mu}$ acts in
$\ell^{2,e}(\mathbb{Z})$ as

$\widehat{V}_{\gamma\lambda\mu}\widehat{f}(x)=\widehat{v}_{\gamma\lambda\mu}(x)\widehat{f}(x).$

###  2.4. The essential spectrum of discrete Schrödinger operators

Since $V_{\gamma\lambda\mu}$ is a finite rank operator, by well known Weyl’s
Theorem (see [34, Theorem XIII.14]) for any $K\in\mathbb{T}$ the essential
spectrum $\sigma_{\mathrm{ess}}(H_{\gamma\lambda\mu}(K))$ of
$H_{\gamma\lambda\mu}$ coincides with the spectrum of $H_{0}(K),$ i.e.,

(2.5)
$\sigma_{\mathrm{ess}}(H_{\gamma\lambda\mu}(K))=\sigma(H_{0}(K))=[{\mathcal{E}}_{\min}(K),{\mathcal{E}}_{\max}(K)],$

where

$\displaystyle{\mathcal{E}}_{\min}(K):=$
$\displaystyle\min_{p\in\mathbb{T}}\,{\mathcal{E}}_{K}(p)=2(1-\cos\tfrac{K}{2})\geq
0={\mathcal{E}}_{\min}(0),$ $\displaystyle{\mathcal{E}}_{\max}(K):=$
$\displaystyle\max_{p\in\mathbb{T}}\,{\mathcal{E}}_{K}(p)=2(1+\cos\tfrac{K}{2})\leq
4={\mathcal{E}}_{\max}(0).$

##  3\. Main results

Let $K\in\mathbb{T}$ and $n_{+}({H}_{\gamma\lambda\mu}(K))$ resp.
$n_{-}({H}_{\gamma\lambda\mu}(K))$ be the number of eigenvalues of the
operator ${H}_{\gamma\lambda\mu}(K)$ above resp. below its essential spectrum.

Our first main result is a generalization of Theorem 2 in [3].

######  Theorem 3.1.

Suppose that ${H}_{\gamma\lambda\mu}(0)$ has $n$ eigenvalues below resp. above
the essential spectrum for some $\gamma,\lambda,\mu\in\mathbb{R}.$ Then for
each $K\in\mathbb{T}$ the operator ${H}_{\gamma\lambda\mu}(K)$ has at least
$n$ eigenvalues below resp. above its essential spectrum. In other words,

$n_{-}({H}_{\gamma\lambda\mu}(K))\geq n_{-}({H}_{\gamma\lambda\mu}(0))$

and

$n_{+}({H}_{\gamma\lambda\mu}(K))\geq n_{+}({H}_{\gamma\lambda\mu}(0)).$

Let us consider the cubic polynomial $C^{\pm}$ of three-variable defined by

(3.1)
$C^{\pm}(\gamma,\lambda,\mu):=\mp(\gamma+\lambda+\mu+\gamma\lambda\mu)+\gamma\lambda+2\gamma\mu+\lambda\mu.$

######  Lemma 3.2.

The set of points $\mathbb{R}^{3}$ satisfying the equation
$C^{\pm}(\gamma,\lambda,\mu)=0$ coincides with the graph of function

$\displaystyle\gamma^{\pm}(\lambda,\mu)=-\frac{Q^{\pm}_{0}(\lambda,\mu)}{Q^{\pm}_{1}(\lambda,\mu)},$

where $Q^{\pm}_{0}(\lambda,\mu)$ and $Q^{\pm}_{1}(\lambda,\mu)$ are defined as

(3.2) $\displaystyle Q^{\pm}_{0}(\lambda,\mu)=\mp(\lambda+\mu)+\lambda\mu,$
$\displaystyle Q^{\pm}_{1}(\lambda,\mu)=\mp(\lambda+2\mu)+1+\lambda\mu.$

###### Proof.

It is sufficient to show that the following system of equations has no
solutions

(3.3) $\begin{cases}Q^{\pm}_{1}(\lambda,\mu)=0\\\
C^{\pm}(\gamma,\lambda,\mu)=0.\end{cases}$

System (3.3) obeys that

(3.4) $\begin{cases}Q^{\pm}_{1}(\lambda,\mu)=0\\\
Q^{\pm}_{0}(\lambda,\mu)=0\end{cases}\quad\Rightarrow\quad\begin{cases}\mu=0\\\
\mu=\pm 1.\end{cases}$

The last system has no solutions. Therefore, (3.3) hasn’t any solutions too.
It leads to the proof of Lemma 3.2. ∎

######  Lemma 3.3.

The set of points of $\mathbb{R}^{2}$ satisfying the equation
$Q^{\pm}_{1}(\lambda,\mu)=0$ coincides with the graph of the function

(3.5) $\displaystyle\lambda^{\pm}(\mu)=-\frac{2\mu\mp 1}{1\mp\mu}.$

###### Proof.

The following system of equations

(3.6) $\begin{cases}1\mp\mu=0\\\ Q^{\pm}_{1}(\lambda,\mu)=0\end{cases}$

is equivalent to (3.4), which has no solutions, thus, it hasn’t any solutions.
It yields the proof of Lemma 3.3. ∎

The straight lines $\mu=-1$ and $\mu=1$ separate the graph of the functions
$\lambda^{-}(\cdot)$ and $\lambda^{+}(\cdot)$ on the $(\lambda,\mu)$-plane
into two (different) continuous curves $\\{\tau^{-}_{1},\tau^{-}_{2}\\}$ and
$\\{\tau^{+}_{1},\tau^{+}_{2}\\}$:

$\displaystyle\tau^{-}_{1}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:\lambda=-\frac{1+2\mu}{1+\mu},\,\,\mu>-1\\},$
$\displaystyle\tau^{-}_{2}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:\lambda=-\frac{1+2\mu}{1+\mu},\,\,\mu<-1\\}$

and

$\displaystyle\tau^{+}_{1}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:\lambda=\frac{1-2\mu}{1-\mu},\,\,\mu<1\\},$
$\displaystyle\tau^{+}_{2}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:\lambda=\frac{1-2\mu}{1-\mu},\,\,\mu>1\\}.$

The curves $\\{\tau^{-}_{1},\tau^{-}_{2}\\}$ and
$\\{\tau^{+}_{1},\tau^{+}_{2}\\}$ divide the $(\lambda,\mu)$\- plane into the
domains $D^{-}_{1},D^{-}_{2},D^{-}_{3}$ and $D^{+}_{1},D^{+}_{2},D^{+}_{3}$ of
the functions $\lambda^{-}(\cdot)$ and $\lambda^{+}(\cdot)$, respectively:

(3.7) $\displaystyle
D^{-}_{1}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{-}_{1}(\lambda,\mu)>0,\,\,\mu>-1\\},$
$\displaystyle
D^{-}_{2}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{-}_{1}(\lambda,\mu)<0\\},$
$\displaystyle
D^{-}_{3}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{-}_{1}(\lambda,\mu)>0,\,\,\mu<-1\\}$

and

(3.8) $\displaystyle
D^{+}_{1}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{+}_{1}(\lambda,\mu)>0,\,\,\mu<1\\},$
$\displaystyle
D^{+}_{2}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{+}_{1}(\lambda,\mu)<0\\},$
$\displaystyle
D^{+}_{3}=\\{(\lambda,\mu)\in\mathbb{R}^{2}:Q^{+}_{1}(\lambda,\mu)>0,\,\,\mu>1\\}.$

Figure 1. The domains $D^{\pm}_{\alpha},\alpha=1,2,3$ of the principle
functions $\gamma^{\pm}$ in the $(\lambda,\mu)$-plane of the parameters
$\lambda,\mu\in\mathbb{R}$.

The following lemma summarizes the locations of the domains
$D^{\pm}_{\alpha},\alpha=1,2,3$ defined in (3.7) and (3.8) and also their
relations.

######  Lemma 3.4.

The followings are true:

  * (i)

$D^{-}_{3}\subseteq D^{+}_{1}$,

  * (ii)

$D^{+}_{3}\subseteq D^{-}_{1}$,

  * (iii)

$D^{-}_{3}\cap D^{+}_{3}=\emptyset$,

  * (iv)

For any $\alpha=1,2,3$ the regions $D^{-}_{\alpha}$ and $D^{+}_{\alpha}$ are
symmetric with respect to origin.

###### Proof.

The definitions of $D^{\pm}_{\alpha},\alpha=1,2,3$ yield the proofs of items
(i)-(iii) of Lemma 3.4 and the equality
$Q^{+}_{1}(\lambda,\mu)=Q^{-}_{1}(-\lambda,-\mu)$ yields (iv) (see Figure 1).
∎

Recall that, the domain of the function $\gamma^{\pm}(\cdot,\cdot)$ is an open
set

$\displaystyle\mathbb{R}^{2}\setminus(\tau^{\pm}_{1}\cup\tau^{\pm}_{2})=D^{\pm}_{1}\cup
D^{\pm}_{2}\cup D^{\pm}_{3}.$

The curves $\tau^{\pm}_{1}$ and $\tau^{\pm}_{2}$ in $\mathbb{R}^{2}$ are
define the corresponding surfaces
$\Upsilon^{\pm}_{j}\subset\mathbb{R}^{3},\,j=1,2$ :

$\displaystyle\Upsilon^{\pm}_{j}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3},\,(\lambda,\mu)\in\tau^{\pm}_{j}\\}.$

Further, the surfaces $\Upsilon^{\pm}_{1}$ and $\Upsilon^{\pm}_{2}$ will
separate the graph of the function $\gamma^{\pm}(\cdot,\cdot)$ into three
different continuous (connected) surfaces $\Gamma^{\pm}_{1}$,
$\Gamma^{\pm}_{2}$ and $\Gamma^{\pm}_{3}$ in $\mathbb{R}^{3}$:

$\displaystyle\Gamma^{\pm}_{j}=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\gamma=-\frac{Q^{\pm}_{0}(\lambda,\mu)}{Q^{\pm}_{1}(\lambda,\mu)},\,\,(\lambda,\mu)\in
D^{\pm}_{j}\\},j=1,2,3.$

The surfaces $\Gamma^{-}_{1},\Gamma^{-}_{2},\Gamma^{-}_{3}$ and
$\Gamma^{+}_{1},\Gamma^{+}_{2},\Gamma^{+}_{3}$ divides the three dimensional
space $\mathbb{R}^{3}$ into four separated connected components
$\mathbb{G}^{-}_{0}$, $\mathbb{G}^{-}_{1}$, $\mathbb{G}^{-}_{2}$,
$\mathbb{G}^{-}_{3}$ and $\mathbb{G}^{+}_{0}$, $\mathbb{G}^{+}_{1}$,
$\mathbb{G}^{+}_{2}$, $\mathbb{G}^{+}_{3}$ respectively:

(3.9)
$\displaystyle\mathbb{G}^{-}_{0}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{-}(\gamma,\lambda,\mu)>0,\,\,(\lambda,\mu)\in
D^{-}_{1}\\},$
$\displaystyle\mathbb{G}^{-}_{1}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{-}(\gamma,\lambda,\mu\\}<0,\,\,(\lambda,\mu)\in\overline{D^{-}_{1}}\cup
D^{-}_{2}\\},$
$\displaystyle\mathbb{G}^{-}_{2}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{-}(\gamma,\lambda,\mu)>0,\,\,(\lambda,\mu)\in\overline{D^{-}_{2}}\cup
D^{-}_{3}\\},$
$\displaystyle\mathbb{G}^{-}_{3}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{-}(\gamma,\lambda,\mu)<0,\,\,(\lambda,\mu)\in
D^{-}_{3}\\}$

and

(3.10)
$\displaystyle\mathbb{G}^{+}_{0}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{+}(\gamma,\lambda,\mu)>0,\,\,(\lambda,\mu)\in
D^{+}_{1}\\},$
$\displaystyle\mathbb{G}^{+}_{1}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{+}(\gamma,\lambda,\mu\\}<0,\,\,(\lambda,\mu)\in\overline{D^{+}_{1}}\cup
D^{+}_{2}\\},$
$\displaystyle\mathbb{G}^{+}_{2}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{+}(\gamma,\lambda,\mu)>0,\,\,(\lambda,\mu)\in\overline{D^{+}_{2}}\cup
D^{+}_{3}\\},$
$\displaystyle\mathbb{G}^{+}_{3}:=\\{(\gamma,\lambda,\mu)\in\mathbb{R}^{3}:\,C^{+}(\gamma,\lambda,\mu)<0,\,\,(\lambda,\mu)\in
D^{+}_{3}\\}.$

######  Theorem 3.5.

Let ${\mathcal{C}}$ be one of the above connected components of the partition
of the $(\gamma,\lambda,\mu)$-space. Then for any
$(\gamma,\lambda,\mu)\in{\mathcal{C}}$ the numbers
$n_{+}({H}_{\gamma\lambda\mu}(0))$ and $n_{-}({H}_{\gamma\lambda\mu}(0))$ of
eigenvalues of $H_{\gamma\lambda\mu}(0)$ lying, respectively, below and above
the essential spectrum (2.5) remain constant.

Now we study the number of eigenvalues of $H_{\gamma\lambda\mu}(0)$ in
$(-\infty,0)$ and $(4,+\infty)$ depending on the potential parameters
$\gamma$, $\lambda$ and $\mu$.

######  Theorem 3.6.

For any $\alpha=0,1,2,3$ the following statements are true:

  * (i)

if $(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{\alpha}$, then
$n_{-}({H}_{\gamma\lambda\mu}(0))=\alpha$;

  * (ii)

if $(\gamma,\lambda,\mu)\in\mathbb{G}^{+}_{\alpha}$, then
$n_{+}({H}_{\gamma\lambda\mu}(0))=\alpha$.

In the following lemma the locations of the connected components

(3.11)
${\mathcal{C}}_{\alpha\beta}=\mathbb{G}_{\alpha}^{-}\cap\mathbb{G}_{\beta}^{+},\,\,\alpha,\beta=0,1,2,3$

are described, which divide the three dimensional space $\mathbb{R}^{3}$ into
a disjoint parts.

######  Lemma 3.7.

The following relations are true:

  * (i)

${\mathcal{C}}_{30}=\mathbb{G}_{3}^{-}\cap\mathbb{G}_{0}^{+}=\mathbb{G}_{3}^{-}$;

  * (ii)

${\mathcal{C}}_{03}=\mathbb{G}_{0}^{-}\cap\mathbb{G}_{3}^{+}=\mathbb{G}_{3}^{+}$;

  * (iii)

If $\alpha+\beta>3$, then
${\mathcal{C}}_{\alpha\beta}=\mathbb{G}_{\alpha}^{-}\cap\mathbb{G}_{\beta}^{+}=\emptyset$.

  * (iv)

If $\alpha+\beta\leq 3$, then ${\mathcal{C}}_{\alpha\beta}$ is symmetric to
${\mathcal{C}}_{\beta\alpha}$ with respect to origin .

###### Proof.

The assertions (i)-(iii) follow from the definition (3.9), Lemma 3.4 and
several simple inequalities. Proof of the assertion (iv) is obtained by the
equality $C^{+}(\gamma,\lambda,\mu)=C^{-}(-\gamma,-\lambda,-\mu)$.

∎

Theorem 3.6 and definition (3.11) of connected components
${\mathcal{C}}_{\alpha\beta}$ yield the following corollary.

######  Corollary 3.8.

Let $\gamma,\lambda,\mu\in\mathbb{R}$. If
$(\gamma,\lambda,\mu)\in{\mathcal{C}}_{\alpha\beta}$ then

$n_{-}({H}_{\gamma\lambda\mu}(0))=\alpha\quad\text{and}\quad
n_{+}({H}_{\gamma\lambda\mu}(0))=\beta,$

where $\alpha,\beta=0,1,2,3$ and $\alpha+\beta\leq 3.$

Recall that by the min-max principle $H_{\gamma\lambda\mu}(K)$ can have at
most three eigenvalues outside its essential spectrum.

The following theorem provides the sharp lower bound for the number of
eigenvalues lying outside the essential spectrum of $H_{\gamma\lambda\mu}(K)$
depending only on $\gamma$,$\lambda$ and $\mu$.

######  Theorem 3.9.

Let $K\in\mathbb{T}$ and $(\gamma,\lambda,\mu)\in\mathbb{R}^{3}.$ For all
$\alpha,\beta=0,1,2,3$ satisfying the condition $\alpha+\beta\leq 3$ the
following relations are true :

  * (i)

$\text{if}\quad(\gamma,\lambda,\mu)\in{\mathcal{C}}_{\alpha\beta}\quad\text{and}\quad\alpha+\beta<3,$
$\text{then}\,\,n_{-}(H_{\gamma\lambda\mu}(K))\geq\alpha,\,\,n_{+}(H_{\gamma\lambda\mu}(K))\geq\beta$;

  * (ii)

$\text{if}\quad(\gamma,\lambda,\mu)\in{\mathcal{C}}_{\alpha\beta}\quad\text{and}\quad\alpha+\beta=3,$
then
$n_{-}(H_{\gamma\lambda\mu}(K))=\alpha,\,\,n_{+}(H_{\gamma\lambda\mu}(K))=\beta.$

##  4\. Proof of the main results

###  4.1. The discrete spectrum of $H_{\gamma\lambda\mu}(0)$

In the case of $K=0$, the Fredholm determinant
$\Delta_{\gamma\lambda\mu}(0,z)$ is easier to study. Note that the essential
spectrum of Hamiltonian $H_{\gamma\lambda\mu}(0)$ coincides with the segment
$[0,4]$.

We try to find an (implicit) equation for the discrete eigenvalues of
$H_{\gamma\lambda\mu}(0)$, i.e., for the non-zero solutions of equation

$H_{\gamma\lambda\mu}(0)f=zf$

in $z\in\mathbb{R}\setminus[0,4]$.

We apply the Fredholm’s determinants method to study the number and location
of eigenvalues (see, e.g., [1, 26]). The Fredholm determinant associated to
$H_{\gamma\lambda\mu}(0)$ can be written as

(4.1) $\Delta_{\gamma\lambda\mu}(0;z)=\left|\begin{array}[]{cc}1+\gamma
a(z)\quad\lambda b(z)\quad\quad\mu c(z)\\\ \quad\gamma b(z)\quad 1+\lambda
d(z)\quad\mu e(z)\\\ \quad\quad\gamma c(z)\quad\lambda e(z)\quad 1+\mu f(z)\\\
\end{array}\right|,$

where

(4.2) $\displaystyle
a(z):=\int\limits_{\mathbb{T}}\frac{2dt}{\varepsilon(t)-z},$ $\displaystyle
b(z):=\int\limits_{\mathbb{T}}\frac{2\cos tdt}{\varepsilon(t)-z},$
$\displaystyle c(z):=\int\limits_{\mathbb{T}}\frac{2\cos
2tdt}{\varepsilon(t)-z},$ $\displaystyle
d(z):=\int\limits_{\mathbb{T}}\frac{2\cos^{2}tdt}{\varepsilon(t)-z},$
$\displaystyle e(z):=\int\limits_{\mathbb{T}}\frac{2\cos t\cos
2tdt}{\varepsilon(t)-z},$ $\displaystyle
f(z):=\int\limits_{\mathbb{T}}\frac{2\cos^{2}{2t}dt}{\varepsilon(t)-z}.$

Functions $a(\cdot)$, $d(\cdot)$ and $f(\cdot)$ are analytic in
$\mathbb{R}\backslash[0,4]$, strictly decreasing in
$\mathbb{R}\backslash[0,\,4]$, positive in the interval $(-\infty,0)$ and
negative in $(4,+\infty)$. The functions $b(\cdot)$, $c(\cdot)$ and $e(\cdot)$
are analytic in $\mathbb{R}\backslash[0,4]$ too. Their behaviour near $z=0$
and $z=4$ are described in the following proposition.

######  Proposition 4.1.

The functions defined in (4.2) have the following asymptotics

$\displaystyle
a(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases},$ $\displaystyle
b(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}-1+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+1+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases},$ $\displaystyle
c(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}-2+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+2+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases},$ $\displaystyle
d(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}-1+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+1+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases},$ $\displaystyle
e(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}-2+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+2+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases},$ $\displaystyle
f(z)=\begin{cases}\frac{1}{(-z)^{\frac{1}{2}}}-2+O(-z)^{\frac{1}{2}},\,\,as\,\,z\nearrow
0\\\ -\frac{1}{(z-4)^{\frac{1}{2}}}+2+O(z-4)^{\frac{1}{2}},\,\,as\,\,z\searrow
4\end{cases}.$

Here $(-z)^{\frac{1}{2}}$ and $(z-4)^{\frac{1}{2}}$ are denote those branches
of analytic functions that are positive for $-z>0$ and $z-4>0$.

Proposition 4.1 can be proved as [21, Proposition 4].

######  Lemma 4.2.

For all $\gamma,\lambda,\mu\in\mathbb{R}$ the determinant
$\Delta_{\gamma\lambda\mu}(0;z)$ has asymptotics

(4.3)
$\displaystyle\Delta_{\gamma\lambda\mu}(0;z)=\begin{cases}C^{-}(\gamma,\lambda,\mu)(-z)^{-\frac{1}{2}}+D^{-}(\gamma,\lambda,\mu)+O((-z)^{\frac{1}{2}}),\,\,as\,\,z\nearrow
0,\\\
C^{+}(\gamma,\lambda,\mu)(z-4)^{-\frac{1}{2}}+D^{+}(\gamma,\lambda,\mu)+O((z-4)^{\frac{1}{2}}),\,\,as\,\,z\searrow
4,\end{cases}$

where $C^{\pm}$ is defined in (3.1) and

$\displaystyle
D^{\pm}(\gamma,\lambda,\mu)=1-(\gamma\lambda+2\lambda\mu+4\gamma\mu)\mp(\lambda+\mu+\gamma\lambda\mu).$

The proof of Lemma 4.2 can be derived by using the asymptotics of functions
$a(\cdot),b(\cdot),c(\cdot),d(\cdot),e(\cdot)$ and $f(\cdot)$ in Proposition
4.1.

######  Corollary 4.3.

For all $(\gamma,\lambda,\mu)\in\mathbb{R}^{3}$ the relations

  * (i).

$\lim\limits_{z\rightarrow{-\infty}}\Delta_{\gamma\lambda\mu}(0;z)=1,$

  * (ii).

$\lim\limits_{z\nearrow
0}\sqrt{-z}\Delta_{\gamma\lambda\mu}(0;z)=C^{-}(\gamma,\lambda,\mu),$

  * (iii).

$\lim\limits_{z\searrow
4}\sqrt{z-4}\Delta_{\gamma\lambda\mu}(0;z)=C^{+}(\gamma,\lambda,\mu)$

hold.

Proof of Corollary 4.3. The first item follows from the Lebesgue dominated
convergence theorem. Lemma 4.2 yields the proof of other items. $\square$

The next lemma describes a one-to-one correspondence between the eigenvalues
$H_{\gamma\lambda\mu}(0)$ and the zeros of the Fredholm determinant
$\Delta_{\gamma\lambda\mu}(0;z)$.

######  Lemma 4.4.

The number $z\in\mathbb{R}\setminus[0,4]$ is an eigenvalue of
$H_{\gamma\lambda\mu}(0)$ if and only if it is a zero of
$\Delta_{\gamma\lambda\mu}(0;\cdot).$ Moreover, in $\mathbb{R}\setminus[0,4]$
the function $\Delta_{\gamma\lambda\mu}(0;\cdot)$ has at most three zeros.

###### Proof.

The first assertion follows from the theory of Fredholm determinants (see, for
example, [1]). Since the operator $H_{\gamma\lambda\mu}(0)$ has rank at most
three, by the minimum-max principle it has at most three eigenvalues outside
the essential spectrum. So, according to the first part of the proposition,
$\Delta_{\gamma\lambda\mu}(0;\cdot)$ has at most three zeros in
$\mathbb{R}\setminus[0,4].$ ∎

The following lemmas determine the number and arrangement of eigenvalues of
the operator $H_{\gamma\lambda 0}(0)$ that lie below the essential spectrum.

######  Lemma 4.5.

  * Let $(\gamma,\lambda)\in\mathbb{R}^{2}$, then the following relations hold.

  * (i)

If $\gamma+\lambda+\gamma\lambda>0$ and $\gamma+1>0$, then $H_{\gamma\lambda
0}(0)$ has no eigenvalues in $(-\infty,0)$.

  * (ii)

If $\gamma+\lambda+\gamma\lambda<0$ or $\gamma+\lambda+\gamma\lambda=0$ and
$\gamma+1>0$, then $H_{\gamma\lambda 0}(0)$ has one eigenvalue in
$(-\infty,0)$.

  * (iii)

If $\gamma+\lambda+\gamma\lambda\geq 0$ and $\gamma+1\leq 0$, then
$H_{\gamma\lambda 0}(0)$ has two eigenvalues in $(-\infty,0)$.

The following lemma provides the dependence of the number of eigenvalues of
the operator $H_{\gamma\lambda 0}(0)$ in $(4,+\infty)$ on $\gamma$ and
$\lambda$:

######  Lemma 4.6.

  * Let $(\gamma,\lambda)\in\mathbb{R}^{2}$, then the following relations hold.

  * (i)

If $-\gamma-\lambda+\gamma\lambda>0$ and $\gamma-1<0$, then $H_{\gamma\lambda
0}(0)$ has no eigenvalues in $(4,+\infty)$.

  * (ii)

If $-\gamma-\lambda+\gamma\lambda<0$ or $-\gamma-\lambda+\gamma\lambda=0$ and
$\gamma-1<0$, then $H_{\gamma\lambda 0}(0)$ has one eigenvalue in
$(4,+\infty)$.

  * (iii)

If $-\gamma-\lambda+\gamma\lambda\geq 0$ and $\gamma-1\geq 0$, then
$H_{\gamma\lambda 0}(0)$ has two eigenvalues in $(4,+\infty)$.

Lemmas 4.5 and 4.6 can be proved as in [25, Theorem 5.5].

Proof of Theorem 3.5. Let us assume, without loss of generality, that
$C^{-}(\gamma,\lambda,\mu)<0$ for all $(\gamma,\lambda,\mu)\in{\mathcal{C}}$.
The definition of determinant and Lemma 4.3 yield

(4.4)
$\lim\limits_{z\to-\infty}\Delta_{\gamma\lambda\mu}(0,z)=1,\,\lim\limits_{z\nearrow
0}\Delta_{\gamma\lambda\mu}(0,z)<0.$

Let $(\gamma_{0},\lambda_{0},\mu_{0})\in{\mathcal{C}}$ be a point of
${\mathcal{C}}$ and $z_{0}<0$ be a zero of the function
$\Delta_{\gamma_{0}\lambda_{0}\mu_{0}}(z)$ of multiplicity $m\geq 1$. For each
fixed $z<0$ the determinant $\Delta_{\gamma\lambda\mu}(z)$ is a real analytic
function in $(\gamma,\lambda,\mu)\in{\mathcal{C}}$ and for each fixed
$\gamma,\lambda,\mu\in\mathbb{R}$ the function $\Delta_{\gamma\lambda\mu}(z)$
is real analytic in $z\in(-\infty,0)$. Hence, for each $\varepsilon>0$ there
are numbers $\delta>0$, $\eta>0$ and an open neighborhood $W_{\eta}(z_{0})$ of
$z_{0}$ with radius $\eta$ such that for all $z\in\overline{W_{\eta}(z_{0})}$
and $(\gamma,\lambda,\mu)\in{\mathcal{C}}$ obeying the conditions
$|z-z_{0}|=\eta$ and
$||(\gamma,\lambda,\mu)-(\gamma_{0},\lambda_{0},\mu_{0})||<\delta$ the
following two inequalities $|\Delta_{\gamma_{0}\lambda_{0}\mu_{0}}(z)|>\eta$
and
$|\Delta_{\gamma\lambda\mu}(z)-\Delta_{\gamma_{0}\lambda_{0}\mu_{0}}(z)|<\varepsilon$
hold. Then by Rouché’s theorem the number of zeros of the function
$\Delta_{\gamma\lambda\mu}(z)$ in ${W_{\eta}(z_{0})}$ remains constant for all
$(\gamma,\lambda,\mu)\in{\mathcal{C}}$ satisfying the inequality
$||(\gamma,\lambda,\mu)-(\gamma_{0},\lambda_{0},\mu_{0})||<\delta$. Since the
root $z_{0}<0$ of the function $\Delta_{\gamma\lambda\mu}(z)$ is arbitrary in
$(B_{1},B_{2})$ we conclude that the number of its zeros remains constant in
$(B_{1},B_{2})$ for all $(\gamma,\lambda,\mu)\in{\mathcal{C}}$ satisfying
$||(\gamma,\lambda,\mu)-(\gamma_{0},\lambda_{0},\mu_{0})||<\delta$.

Further each Jordan curve $\Gamma\subset{\mathcal{C}}$ connecting any two
points of ${\mathcal{C}}$ is a compact set, so the number of zeros of the
function $\Delta_{\gamma\lambda\mu}(z)$ lying below the bottom of the
essential spectrum for any $(\gamma,\lambda,\mu)\in\Gamma$ remains constant.
Therefore, Lemma 4.4 yields that the number of eigenvalues
$n_{-}\bigl{(}H_{\gamma\lambda\mu}(0)\bigr{)}$ of the operator
${H}_{\gamma\lambda\mu}(0)$ below the essential spectrum is constant.

The proof in the case of $n_{+}\bigl{(}H_{\gamma\lambda\mu}(0)\bigr{)}$ is
done in the same way. $\square$

Proof of Theorem 3.6. Let us prove Theorem 3.6 in the cases $\alpha=0,1,2,3$
successively. According to (3.9) and (3.1) we have
$(0,0,1)\in\mathbb{G}^{-}_{0}$.The representation (4.1) of the determinant
$\Delta_{\gamma\lambda\mu}(0;z)$ yields that

$\Delta_{001}(0;z)=1+f(z).$

By definition (4.2) of $f(z)$, for all $z\in(-\infty,0)$ the inequalities
$f(z)>0$ and $\Delta_{001}(0;z)>0$ hold, i.e., the determinant
$\Delta_{001}(0;z)$ has no negative zeros in $z\in(-\infty,0)$. Lemma 4.4
yields that the operator $H_{001}(0)$ has no eigenvalues below the essential
spectrum. Theorem 3.5 gives that the operator $H_{\gamma\lambda\mu}(0)$ has no
eigenvalues below the essential spectrum for all
$(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{0}$.

Due to (3.9) we have $(0,-1,0)\in\mathbb{G}^{-}_{1}$. Lemma 4.5 gives that the
operator $H_{0(-1)0}(0)$ has exactly one eigenvalue. Theorem 3.5 yields that
for any $(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{1}$, the operator
$H_{\gamma\lambda\mu}(0)$ has exactly one eigenvalue below the essential
spectrum.

We note $(-3,-3,0)\in\mathbb{G}^{-}_{2}$. According to (iii) of Lemma 4.5 the
operator $H_{(-3)(-3)0}(0)$ has two eigenvalues. Theorem 3.5 yields that for
any $(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{2}$ the operator
$H_{\gamma\lambda\mu}(0)$ has two eigenvalues below the essential spectrum.

Now assume that $(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{3}$.

Due to (3.9) we have

$C^{-}(\gamma,\lambda,\mu)<0,\,\,(\lambda,\mu)\in D^{-}_{3}.$

Definition 3.7 and inclusion $(\lambda,\mu)\in D^{-}_{3}$ give

(4.5) $\mu<-1,\,\,Q_{1}^{-}(\lambda,\mu)>0.$

Inequalities (4.5) and $C^{-}(\gamma,\lambda,\mu)<0$ guarantee that

(4.6)
$\gamma<-\frac{Q_{0}^{-}(\lambda,\mu)}{Q_{1}^{-}(\lambda,\mu)}=\frac{\mu+1}{Q_{1}^{-}(\lambda,\mu)}-1<0.$

According to negativity of $\gamma$ the function $\Delta_{\gamma
00}(0;z)=1+\gamma a(z)$ is continuous and monotone decreasing in
$(-\infty,0)$.
By (3.1) and (4.6) we have that $C^{-}(\gamma,0,0)=\gamma<0.$ Corollary 4.3
yields that

(4.7) $\quad\lim\limits_{z\rightarrow-\infty}\Delta_{\gamma
00}(0;z)=1\quad\text{and}\lim\limits_{z\nearrow 0}\Delta_{\gamma
00}(0;z)=-\infty.$

Since the function $\Delta_{\gamma 00}(0;z)$ is continuous and monotone
decreasing in the interval $(-\infty,0)$ it has exactly one zero $z_{11}$
below the essential spectrum. Obviously

(4.8) $\displaystyle\Delta_{\gamma 00}(0;z)=1+\gamma a(z)>0\quad\text{if}\quad
z<z_{11},$ $\displaystyle\Delta_{\gamma 00}(0;z)=1+\gamma
a(z)<0\quad\text{if}\quad z>z_{11}.$

Observe

(4.9) $\Delta_{\gamma\lambda 0}(0;z_{11})=(1+\gamma a(z_{11}))(1+\lambda
d(z_{11}))-\gamma\lambda(b(z_{11}))^{2}=-\gamma\lambda(b(z_{11}))^{2}<0.$

According to inequalities (4.5) we have

$C^{-}(\gamma,\lambda,0)=\gamma+\lambda+\gamma\lambda=Q_{1}(\lambda,\mu)-(\mu+1)>0.$

Corollary 4.3 and inequality (4.9) give

$\lim\limits_{z\rightarrow-\infty}\Delta_{\gamma\lambda
0}(0;z)=1,\quad\Delta_{\gamma\lambda
0}(0;z_{11})<0\quad\text{and}\quad\lim\limits_{z\nearrow
0}\Delta_{\gamma\lambda 0}(0;z)=+\infty.$

Thus, the continuous function $\Delta_{\gamma\lambda 0}(0;z)$ has at least one
zero in each intervals $(-\infty,z_{11})$ and $(z_{11},0)$. Therefore there
exists real numbers satisfying the inequalities

(4.10) $z_{21}<z_{11}<z_{22}<0$

such that the following equalities hold:

$\Delta_{\gamma\lambda 0}(0;z_{21})=\Delta_{\gamma\lambda 0}(0;z_{22})=0.$

Lemma 4.4 and the min-max principle yield that $H_{\gamma\lambda 0}(0)$ has at
least two eigenvalues below the essential spectrum. Hence the determinant
$\Delta_{\gamma\lambda 0}(0;z)$ has exactly two zeros $z_{21}$ and $z_{22}$ in
$(-\infty,0)$, which yields

(4.11) $\displaystyle(1+\gamma a(z_{21}))(1+\lambda
d(z_{21}))=\gamma\lambda(b(z_{21}))^{2}>0,$ $\displaystyle(1+\gamma
a(z_{22}))(1+\lambda d(z_{22}))=\gamma\lambda(b(z_{22}))^{2}>0.$

The relations (4.8), (4.10) and (4.11) yield

$\displaystyle 1+\gamma a(z_{21})>0\quad\text{and}\quad 1+\lambda
d(z_{21})>0,$ $\displaystyle 1+\gamma a(z_{22})<0\quad\text{and}\quad
1+\lambda d(z_{22})<0.$

We note that $(\gamma,\lambda,\mu)\in\mathbb{G}^{-}_{3}$ and so
$C^{-}(\gamma,\lambda,\mu)>0$. Applying Corollary 4.3 we have that

$\lim\limits_{z\rightarrow-\infty}\Delta_{\gamma\lambda\mu}(0;z)=1\quad\text{and}\quad\lim\limits_{z\nearrow
0}\Delta_{\gamma\lambda\mu}(0;z)=-\infty.$

Observe that

$\displaystyle\Delta_{\gamma\lambda\mu}(0;z_{21})=$
$\displaystyle\Delta_{\gamma\lambda 0}(0;z_{21})[1+\mu
f(z_{21})]+2\gamma\lambda\mu b(z_{21})c(z_{21})e(z_{21})-$
$\displaystyle\gamma\mu[1+\lambda d(z_{21})]c^{2}(z_{21})-\lambda\mu[1+\gamma
a(z_{21})](e(z_{21}))^{2}=$ $\displaystyle\mu\big{[}\sqrt{-\gamma[1+\lambda
d(z_{21})]}c(z_{21})+\sqrt{-\lambda[1+\gamma
a(z_{21})]}e(z_{21})\big{]}^{2}<0.$

Similarly can be shown

$\displaystyle\Delta_{\gamma\lambda\mu}(0;z_{22})=-\mu\big{[}\sqrt{\gamma[1+\lambda
d(z_{22})]}c(z_{22})+\sqrt{\lambda[1+\gamma
a(z_{22})]}e(z_{22})\big{]}^{2}>0.$

Then the relations

$\lim\limits_{z\rightarrow-\infty}\Delta_{\gamma\lambda\mu}(0;z)=1,\quad\Delta_{\gamma\lambda\mu}(0;z_{21})<0,\quad\Delta_{\gamma\lambda\mu}(0;z_{22})>0,\quad\lim\limits_{z\nearrow
0}\Delta_{\gamma\lambda\mu}(0;z)=-\infty.$

yield the existence of zeros $z_{31}$, $z_{32}$ and $z_{33}$ of function
$\Delta_{\gamma\lambda\mu}(0;z)$ satisfying the inequalities
$z_{31}<z_{21}<z_{32}<z_{22}<z_{33}<0$ , i.e

$\Delta_{\gamma\lambda\mu}(0;z_{31})=\Delta_{\gamma\lambda\mu}(0;z_{32})=\Delta_{\gamma\lambda\mu}(0;z_{33})=0.$

Hence the function $\Delta_{\gamma\lambda\mu}(0;z)$ has three single zeros
less than $0$. Lemma 4.4 gives that the operator $H_{\gamma\lambda\mu}(0)$ has
three eigenvalues below the essential spectrum.

The proof of item (ii) of the theorem is carried out similarly to the proof of
item (i). $\square$

###  4.2. The discrete spectrum of $H_{\gamma\lambda\mu}(K)$

For every $n\geq 1$ define

(4.12)
$e_{n}(K;\gamma,\lambda,\mu):=\sup\limits_{\phi_{1},\ldots,\phi_{n-1}\in
L^{2}(\mathbb{T})}\,\,\inf\limits_{{\psi}\in[\phi_{1},\ldots,\phi_{n-1}]^{\perp},\,\|{\psi}\|=1}({H}_{\gamma\lambda\mu}(K){\psi},{\psi})$

and

(4.13)
$E_{n}(K;\gamma,\lambda,\mu):=\inf\limits_{\phi_{1},\ldots,\phi_{n-1}\in
L^{2}(\mathbb{T})}\,\,\sup\limits_{{\psi}\in[\phi_{1},\ldots,\phi_{n-1}]^{\perp},\,\|{\psi}\|=1}({H}_{\gamma\lambda\mu}(K){\psi},{\psi}).$

By the minimax principle,
$e_{n}(K;\gamma,\lambda,\mu)\leq{\mathcal{E}}_{\min}(K)$ and
$E_{n}(K;\gamma,\lambda,\mu)\geq{\mathcal{E}}_{\max}(K).$ Since, the rank of
$V_{\gamma\lambda\mu}$ does not exceed three, by choosing $\phi_{1}\equiv 1,$
$\phi_{2}(p)=\cos p$ and $\phi_{3}(p)=\cos 2p$ we immediately see that
$e_{n}(K;\gamma,\lambda,\mu)={\mathcal{E}}_{\min}(K)$ and
$E_{n}(K;\gamma,\lambda,\mu)={\mathcal{E}}_{\max}(K)$ for all $n\geq 4.$

######  Lemma 4.7.

Let $n\geq 1$. Then the maps

$K\in\mathbb{T}\mapsto{\mathcal{E}}_{\min}(K)-e_{n}(K;\gamma,\lambda,\mu)$

and

$K\in\mathbb{T}\mapsto E_{n}(K;\gamma,\lambda,\mu)-{\mathcal{E}}_{\max}(K)$

is non-increasing in $(-\pi,0]$ and non-decreasing in $[0,\pi]$.

###### Proof.

For $\psi\in L^{2}(\mathbb{T})$, we consider

$((H_{0}(K)-{\mathcal{E}}_{\min}(K))\psi,\psi)=\int_{\mathbb{T}}2\cos\tfrac{K}{2}\,\big{(}1-\cos
q\big{)}|\psi(q)|^{2}\,\mathrm{d}q.$

Then, the map
$K\in\mathbb{T}\mapsto((H_{0}(K)-{\mathcal{E}}_{\min}(K))\psi,\psi)$ is non-
decreasing in $(-\pi,0]$ and is non-increasing in $[0,\pi].$ Since
$V_{\gamma\lambda\mu}$ is independent of $K,$ from the definition of
$e_{n}(K;\gamma,\lambda,\mu)$, the map $K\in\mathbb{T}\mapsto
e_{n}(K;\gamma,\lambda,\mu)-{\mathcal{E}}_{\min}(K)$ is non-decreasing in
$(-\pi,0]$ and is non-increasing in $[0,\pi].$ ∎

Proof of Theorem 3.1. For any for any $K\in\mathbb{T}$ and $m\geq 1$ Lemma 4.7
gives

(4.14) $\displaystyle
0\leq{\mathcal{E}}_{\min}(0)-e_{m}(0;\gamma,\lambda,\mu)\leq{\mathcal{E}}_{\min}(K)-e_{m}(K;\gamma,\lambda,\mu),$
and (4.15) $\displaystyle 0\leq
E_{m}(0;\gamma,\lambda,\mu)-{\mathcal{E}}_{\max}(0)\leq
E_{m}(K;\gamma,\lambda,\mu)-{\mathcal{E}}_{\max}(K).$

By assumption $e_{n}(0;\gamma,\lambda,\mu)$ is an discrete eigenvalue of
$H_{\gamma\lambda\mu}(0)$ lying below the bottom ${\mathcal{E}}_{\min}(K)$. So
${\mathcal{E}}_{\min}(0)-e_{n}(0;\gamma,\lambda,\mu)>0$ and hence, by (4.14)
and (2.5), $e_{n}(K;\gamma,\lambda,\mu)$ is a discrete eigenvalue of
$H_{\gamma\lambda\mu}(K)$ for any $K\in\mathbb{T}.$ Since
$e_{1}(K;\gamma,\lambda,\mu)\leq\ldots\leq
e_{n}(K;\gamma,\lambda,\mu)<{\mathcal{E}}_{\min}(K),$ it follows that
$H_{\gamma\lambda\mu}(K)$ has at least $n$ eigenvalue below the essential
spectrum. The case of $E_{n}(K;\gamma,\lambda,\mu)$ is similar. $\square$

Proof of Theorem 3.9 is obtained by combining Theorem 3.1 and Corollary 3.8.
$\square$

Acknowledgments. The authors acknowledge support of this research by Ministry
of Innovative Development of the Republic of Uzbekistan (Grant No.
FZ–20200929224).

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