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Four-Electron Systems in the Impurity Hubbard Model. Second Triplet State. Spectra of the System in the ν-Dimensional Lattice Zν

Abstract

We consider an energy operator of four-electron system in the Impurity Hubbard model with a coupling between nearest-neighbors. The spectrum of the systems in the second triplet state in a ν-dimensional lattice is investigated. For investigation the structure of essential spectra and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model, for which the momentum representation is convenient. In addition, we used the tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces and described the structure of essential spectrum and discrete spectrum of the energy operator of four-electron systems in an impurity Hubbard model for the second triplet state of the system. The investigations show that the essential spectrum of the system consists of the union of no more than sixteen segments, and the discrete spectrum of the system consists of no more than eleven eigenvalues.

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Tashpulatov, S. and Parmanova, R. (2023) Four-Electron Systems in the Impurity Hubbard Model. Second Triplet State. Spectra of the System in the ν-Dimensional Lattice Zν. Journal of Applied Mathematics and Physics, 11, 3393-3427. doi: 10.4236/jamp.2023.1111217.

1. Introduction

The Hubbard model first appeared in 1963 in the works [1] [2] [3] . The model proposed in [1] [2] [3] was called the Hubbard model after John Hubbard, who made a fundamental contribution to studying the statistical mechanics of that system, although the local form of Coulomb interaction was first introduced for an impurity model in a metal by Anderson [4] . We also recall that the Hubbard model is a particular case of the Shubin-Wonsowsky polaron model [5] , which had appeared 30 years before [1] [2] [3] . In the Shubin-Wonsowsky model, along with the on-site Coulomb interaction, the interaction of electrons on neighboring sites is also taken into account.

The Hubbard model well describes the behavior of particles in a periodic potential at sufficiently low temperatures such that all particles are in the lower Bloch band and long-range interactions can be neglected. If the interaction between particles on different sites is taken into account, then the model is often called the extended Hubbard model. It was proposed for describing electrons in solids, and it remains especially interesting since then for studying high-temperature superconductivity. Later, the extended Hubbard model also found applications in describing the behavior of ultracold atoms in optical lattices. In considering electrons in solids, the Hubbard model can be considered a sophisticated version of the model of strongly bound electrons, involving only the electron hopping term in the Hamiltonian. In the case of strong interactions, these two models can give essentially different results. The Hubbard model exactly predicts the existence of so-called Mott insulators, where conductance is absent due to strong repulsion between particles. The Hubbard model is based on the approximation of strongly coupled electrons. In the strongcoupling approximation, electrons initially occupy orbital’s in atoms (lattice sites) and then hop over to other atoms, thus conducting the current. Mathematically, this is represented by the so-called hopping integral. This process can be considered the physical phenomenon underlying the occurrence of electron bands in crystal materials. But the interaction between electrons is not considered in more general band theories. In addition to the hopping integral, which explains the conductance of the material, the Hubbard model contains the so-called on-site repulsion, corresponding to the Coulomb repulsion between electrons. This leads to a competition between the hopping integral, which depends on the mutual position of lattice sites, and the on-site repulsion, which is independent of the atom positions. As a result, the Hubbard model explains the metal-insulator transition in oxides of some transition metals. When such a material is heated, the distance between nearest-neighbor sites increases, the hopping integral decreases, and on-site repulsion becomes dominant.

The Hubbard model is currently one of the most extensively studied multielectron models of metals [6] - [12] and [13] , chapter III, PP. 75-191. In the review [7] , the results obtained on the Hubbard model are summarized. According to the Hubbard model, the more progress that is made in obtaining theoretical solutions, the clearer it becomes that this simple model can exhibit a startling array of phases and regimes, many of which have clear parallels with observed behaviors of a wide variety of complex materials.

For instance, there is compelling evidence that ferromagnetism, various forms of antiferromagnetism, unconventional superconductivity, charge-density waves, electronic liquid crystalline phases, and topologically ordered phases (e.g., “spin liquids”), among other phases, occur in specific realizations of the Hubbard model.

It is our purpose here to summarize, to the extent possible in a brief article, what is established concerning the quantum phases of the Hubbard model. The role of the Hubbard model, which it played in the study of high-temperature superconductivity in cuprates, is discussed.

It is shown that the positive eigenvalues in the Hubbard model (corresponding to repulsive effectual interactions) weaken, and the negative ones grow. The various eigenfunctions correspond to, but are not completely determined by, an irreducible representation of a group of crystal points in the Hubbard model.

Obtaining exact results for the spectrum and wave functions of the crystal described by the Hubbard model and impurity Hubbard model is of great interest.

The spectrum and wave functions of the system of two electrons in a crystal described by the Hubbard Hamiltonian were studied in [14] . It is known that two-electron systems can be in two states, triplet and singlet [14] . In the work [14] is considered the Hamiltonian of the form

H=Am,γa+m,γam,γ+Bm,τ,γa+m,γam+τ,γ+Uma+m,am,a+m,am,. (1)

Here A is the electron energy at a lattice site, B is the transfer integral between neighboring sites, τ=±ej,j=1,2,,ν , where ej are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U is the parameter of the on-site Coulomb interaction of two electrons, γ is the

spin index, γ= or γ= , and denote the spin values 12 and 12 ,

and a+m,γ and am,γ are the respective electron creation and annihilation operators at a site mZν . It was proved in [14] that the spectrum of the system Hamiltonian Ht in the triplet state is purely continuous and coincides with a segment [m,M]=[2A4Bν,2A+4Bν] , where ν is the lattice dimensionality, and the operator Hs of the system in the singlet state, in addition to the continuous spectrum [m,M] , has a unique antibound state for some values of the quasimomentum. For the antibound state, correlated motion of the electrons is realized under which the contribution of binary states is large. Because the system is closed, the energy must remain constant and large. This prevents the electrons from being separated by long distances. Next, an essential point is that bound states (sometimes called scattering-type states) do not form below the continuous spectrum. This can be easily understood because the interaction is repulsive. We note that a converse situation is realized for U<0 : below the continuous spectrum, there is a bound state (antibound states are absent) because the electrons are then attracted to one another.

For the first band, the spectrum is independent of the parameter U of the on-site Coulomb interaction of two electrons and corresponds to the energy of two noninteracting electrons, being exactly equal to the triplet band. The second band is determined by Coulomb interaction to a much greater degree: both the amplitudes and the energy of two electrons depend on U, and the band itself disappears as U0 and increases without bound as U . The second band largely corresponds to a one-particle state, namely, the motion of the doublet, i.e., two-electron bound states.

The spectrum and wave functions of the system of three electrons in a crystal described by the Hubbard Hamiltonian were studied in [15] . In the three-electron systems are exists quartet state, and two type doublet states.

The spectrum of the energy operator of system of four electrons in a crystal described by the Hubbard Hamiltonian in the triplet state was studied in [16] . In the four-electron systems are exists quintet state, and three type triplet states, and two type singlet states. The spectrum of the energy operator of four-electron systems in the Hubbard model in the quintet, and singlet states were studied in [17] .

In the work [18] is considered dominant correlation effects in two-electron atoms.

The use of films in various areas of physics and technology arouses great interest in studying a localized impurity state (LIS) of magnet. Therefore, it is important to study the spectral properties of electron systems in the impurity Hubbard model.

The spectrum and wave functions of the system of two electrons in a crystal described by the impurity Hubbard Hamiltonian were studied in [19] and [20] .

The spectrum of the energy operator of three-electron systems in the impurity Hubbard model in the second doublet state was studied [21] . The structure of essential spectra and discrete spectrum of three-electron systems in the impurity Hubbard model in the Quartet state were studied in [22] . The structure of essential spectra and discrete spectrum of four-electron systems in the impurity Hubbard model in the Quartet state and in the first triplet state were studied in [23] and [24] in the one-dimensional lattice.

In this paper we give a full description of the structure of the essential spectra and discrete spectrum of four-electron systems in the impurity Hubbard model for second triplet state. In contrast to the works [23] and [24] , not only the one-dimensional case is considered here, but the cases when ν=1,2,3 and the spectrum of the system for the second triplet state is described for all values of the parameters of the Hamiltonian. First, using the standard anticommutation relations between the operators of electron creation and annihilation at the lattice sites, we get a coordinate representation of the Hamiltonian action, and then moving on to the Fourier transformation we get a quasi-pulse representation of the Hamiltonian action. Using the concept of tensor products of Hilbert spaces, and tensor products of operators in Hilbert spaces, we bring the problems of studying the spectrum of the energy operator of four electron systems in the Impurity Hubbard model to the study of the spectrum of the energy operator of one electron system in the Impurity Hubbard model. Then, using the results obtained from the study of the spectrum of the energy operator of one-electron systems in the impurity Hubbard model, we describe the spectrum of four electron systems in the Impurity Hubbard model for the second triplet state. The results obtained show how the results of this work differ from the results of the works [23] and [24] . The main result of this paper is Theorems 8 and 9, which describe the spectrum of considered model for second triplet state. The results of sections 2 and 3 and 4 (Theorem 7) there are preliminary facts for the proof of Theorems 8 and 9.

2. Preliminaries

We consider the energy operator of four-electron systems in the Impurity Hubbard model and describe the structure of the essential spectra and discrete spectrum of the system for second triplet state in the lattice. The Hamiltonian of the chosen model has the form

H=Am,γa+m,γam,γ+Bm,τ,γa+m,γam+τ,γ+Uma+m,am,a+m,am,+(A0A)γa+0,γa0,γ+(B0B)τ,γ(a+0,γaτ,γ+a+τ,γa0,γ)+(U0U)a+0,a0,a+0,a0,. (2)

Here A (A0) is the electron energy at a regular (impurity) lattice site; B>0 ( B0>0 ) is the transfer integral between electrons (between electron and impurity) in a neighboring sites, τ=±ej,j=1,2,,ν , where ej are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U (U0) is the parameter of the on-site Coulomb interaction of two electrons, correspondingly in the regular (impurity) lattice site; γ is the spin

index, γ= or γ= , and denote the spin values 12 and 12 , and

a+m,γ and am,γ are the respective electron creation and annihilation operators at a site mZν . The second triplet state corresponds four-electron bound states (or antibound states) to the basis functions: 2t1p,q,r,kZν=a+p,a+q,a+r,a+k,φ0 . The subspace ˜H21t , corresponding to the second triplet state is the set of all vectors of the form ψ21t=p,q,r,kZνf(p,q,r,k)t21p,q,r,kZν , flas2 , where las2 is the subspace of antisymmetric functions in l2((Zν)4) . In this case, the Hamiltonian H acts in the antisymmetric Fock space ˜H21t . Let φ0 be the vacuum

vector in the antisymmetrical Fock space ˜H21t . Let ˜H21t be the restriction H to the subspace ˜H21t . The second triplet state corresponds the free motions of four-electrons in the lattice and their interactions. Let ε1=A0A , ε2=B0B , ε3=U0U .

The energy of the system depends on its total spin S. Along with the Hamiltonian, the Ne electron system is characterized by the total spin S, S=Smax,Smax1,,Smin , Smax=Ne2 , Smin=0,12 .

Hamiltonian (2) commutes with all components of the total spin operator S=(S+,S,Sz) , and the structure of eigenfunctions and eigenvalues of the system therefore depends on S. The Hamiltonian H acts in the antisymmetric Fo’ck space Has .

Below we give the constructions of the Fo’ck space F(H) .

Let H be a Hilbert space and denote by Hn the n- fold tensor product Hn=HHH . We set H0=C and F(H)=n=0Hn . The F(H) is called the Fo’ck space over H ; it will be separably, if H is. For example, if H=L2(R) , then an element ψF(H) is a sequence of functions ψ={ψ0,ψ1(x1),ψ2(x1,x2),ψ3(x1,x2,x3),} , so that |ψ0|2+n=1Rn|ψn(x1,x2,,xn)|2dx1dx2dxn< . Actually, it is not F(H) , itself, but two of its subspaces which are used most frequently in quantum field theory. These two subspaces are constructed as follows: Let Pn be the permutation group on n elements, and let {ψn} be a basis for space H . For each σPn , we define an operator (which we also denote by σ ) on basis elements Hn , by σ(φk1φk2φkn)=φkσ(1)φkσ(2)φkσ(n) . The operator σ extends by linearity to a bounded operator (of norm one) on space Hn , so we can define Sn=1n!σPnσ . That the operator Sn is the operator of orthogonal projection: S2n=Sn , and Sn=Sn . The range of Sn is called n- fold symmetric tensor product of H . In the case, where H=L2(R) and Hn=L2(R)L2(R)L2(R)=L2(Rn) , SnHn is just the subspace of L2(Rn) , of all functions, left invariant under any permutation of the variables. We now define Fs(H)=n=0SnHn . The space Fs(H) is called the symmetrical Fo’ck space over H , or Boson Fo’ck space over H .

Let ε(.) is function from Pn to {1,1} , which is one on even permutations and minus one on odd permutations. Define An=1n!σPnε(σ)σ ; then An is an orthogonal projector on Hn . AnHn is called the n- fold antisymmetrical tensor product of H . In the case where H=L2(R) , AnHn is just the subspace of L2(Rn) , consisting of those functions odd under interchange of two coordinates. The subspace Fa(H)=n=0AnHn is called the antisymmetrical Fo’ck space over H , or the Fermion Fo’ck space over H .

Let φ0 be the vacuum vector in the antisymmetrical Fock space ˜H21t . Let ˜H21t be the restriction H to the subspace ˜H21t .

Theorem 1. The subspace H21t is invariant under the operator H, and the restriction H21t of operator H to the subspace H21t is a bounded self-adjoint operator. It generates a bounded self-adjoint operator ˉH21t acting in the space las2 as

ˉH21tψ21t=4Af(p,q,r,k)+Bτ[f(p+τ,q,r,k)+f(p,q+τ,r,k)+f(p,q,r+τ,k)+f(p,q,r,k+τ)]+U[δp,r+δq,r+δr,k]f(p,q,r,k)

+(A0A)(δp,0+δq,0+δr,0+δk,0)f(p,q,r,k)+(B0B)τ[δp,0f(τ,q,r,k)+δq,0f(p,τ,r,k)+δr,0f(p,q,τ,k)+δk,0f(p,q,r,τ)+δp,τf(0,q,r,k)+δq,τf(p,0,r,k)+δr,τf(p,q,0,k)+δk,τf(p,q,r,0)]+(U0U)[δp,rδp,0+δq,rδq,0+δr,kδr,0]f(p,q,r,k). (3)

The operator H21t acts on a vector ψ21tH21t as

H21tψ21t=p,q,r,kZν(ˉH21tf)(p,q,r,k)2t1p,q,r,kZν. (4)

Proof. We act with the Hamiltonian H on vectors ψ21tH21t using the standard anticommutation relations between electron creation and annihilation operators at lattice sites, {am,γ,a+n,β}=δm,nδγ,β , {am,γ,an,β}={a+m,γ,a+n,β}=θ , and also take into account that am,γφ0=θ , where θ is the zero element of H21t . This yields the statement of the theorem.

Lemma 1. The spectra of the operators H21t and ˉH21t coincide.

Proof. Because the operators H21t and ˉH21t are bounded self-adjoint operators, it follows that if λσ(H21t) , then the Weyl criterion (see [25] , chapter VII, paragraph 3, pp. 262-263) implies that there is a sequence {ψi}i=1 such that ψi=1 and limi(H21tλ)ψi=0 . We set

ψi=p,q,r,kfi(p,q,r,k)a+p,a+q,a+r,a+k,φ0 . Then

(H21tλ)ψi2=((H21tλ)ψi,(2H1tλ)ψi)=p,q,r,k(ˉH21tλ)fi(p,q,r,k)2(a+p,a+q,a+r,a+k,φ0,a+p,a+q,a+r,a+k,φ0)=p,q,r,k(ˉH21tλ)Fi(p,q,r,k)2(ak,ar,aq,ap,a+p,a+q,a+r,a+k,φ0,φ0)=p,q,r,k(ˉH21tλ)Fi(p,q,r,k)2(φ0,φ0)=p,q,r,k(ˉH21tλ)Fi(p,q,r,k)20

as i , where Fi=p,q,r,kfi(p,q,r,k) . It follows that λσ(ˉH21t) . Consequently, σ(H21t)σ(ˉH21t) .

Conversely, let ˉλσ(ˉH21t) . Then, by the Weyl criterion, there is a sequence {Fi}i=1 such that Fi=1 and limi(ˉH21tˉλ)ψi=0 . Setting Fi=p,q,r,kfi(p,q,r,k) , Fi=(p,q,r,k|fi(p,q,r,k)|2)12 , we conclude that ψi=Fi=1 and (ˉH21tˉλ)Fi=(ˉH11tˉλ)ψi0 as i . This means that ˉλσ(H21t) and hence σ(ˉH21t)σ(H21t) . These two relations imply σ(H21t)=σ(ˉH21t) .

We call the operator H21t the four-electron second triplet state operator in the impurity Hubbard model.

Let F:l2((Zν)4)L2((Tν)4)˜H21t be the Fourier transform, where Tν is the ν-dimensional torus endowed with the normalized Lebesgue measure dλ , i.e. λ(Tν)=1 .

We set ˜H21t=FˉH21tF1 . In the quasimomentum representation, the operator ˉH21t acts in the Hilbert space Las2((Tν)4) , where Las2 is the subspace of antisymmetric functions in L2((Tν)4) .

Theorem 2. The Fourier transform of operator ˉH21t is an operator ˜H21t=FˉH21tF1 acting in the space ˜H21t be the formula

˜H21tψ21t=h(λ,μ,γ,θ)f(λ,μ,γ,θ)+UTν[f(s,μ,λ+γs,θ)+f(λ,s,μ+γs,θ)+f(λ,μ,s,γ+θs)]ds+(A0A)[Tνf(s,μ,γ,θ)ds+Tνf(λ,l,γ,θ)dl+Tνf(λ,μ,ξ,θ)dξ+Tνf(λ,μ,γ,n)dn]+(B0B)×[νj=1Tν2[cosλj+cossj]f(s,μ,γ,θ)ds+νj=1Tν2[cosμj+coslj]

×f(λ,l,γ,θ)dl+νj=1Tν2[cosγj+cosξj]f(λ,μ,ξ,θ)dξ+νj=1Tν2[cosθj+cosnj]f(λ,μ,γ,n)dn]+(U0U)[TνTνf(s,μ,ξ,θ)dsdξ+TνTνf(λ,l,ξ,θ)dldξ+TνTνf(λ,μ,ξ,n)dξdn], (5)

where h(λ,μ,γ,θ)=4A+2Bνj=1[cosλj+cosμj+cosγj+cosθj] .

The proof Theorem 2, is straightforward of (3) using the Fourier transformation.

The spectral properties of four-electron systems in the impurity Hubbard model in the second triplet state are closely related to those of its one-electron subsystems in the impurity Hubbard model. Therefore we first study the spectrum and localized impurity states of one-electron impurity systems.

3. One-Electron Impurity Systems

The Hamiltonian of one-electron impurity system has the form:

H=Am,γa+m,γam,γ+Bm,τ,γa+m,γam+τ,γ+(A0A)γa+0,γa0,γ+(B0B)τ,γ(a+0,γaτ,γ+a+τ,γa0,γ), (6)

here A (A0) is the electron energy at a regular (impurity) lattice site; B>0 ( B0>0 ) is the transfer integral between electrons (between electron and impurity)in a neighboring sites, τ=±ej,j=1,2,,ν , where ej are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors; γ is the spin index, γ= or γ= , and denote the spin

values 12 and 12 , and a+m,γ and am,γ are the respective electron creation and annihilation operators at a site mZν .

We let H1 denote the Hilbert space spanned by the vectors in the form ψ=pa+p,φ0 . It is called the space of one-electron states of the operator H. The space H1 is invariant with respect to action of the operator H. Denote by H1 the restriction of H to the subspace H1 .

As in the proof of Theorem 1, using the standard anticommutation relations between electron creation and annihilation operators at lattice sites, we get the following

Theorem 3. The subspace H1 is invariant with respect to the action of the operator H, and the operator H1 is a linear bounded self-adjoint operator, acting in H1 as

H1ψ=p(ˉH1f)(p)a+p,φ0,ψH1, (7)

where ˉH1 is a linear bounded self-adjoint operator acting in the space l2 as

(ˉH1f)(p)=Af(p)+Bτf(p+τ)+ε1δp,0f(p)+ε2τ(δp,τf(0)+δp,0f(τ)). (8)

Lemma 2. The spectra of the operators ˉH1 and H1 coincide.

The proof of Lemma 2 is the same as the proof of the Lemma 1.

As in section 2 denote by F:l2(Zν)L2(Tν)˜H1 the Fourier transform. Setting ˜H1=FˉH1F1 we get that the operator ˉH1 acts in the Hilbert space L2(Tν) .

Using the equality (8) and properties of the Fourier transform we have the following

Theorem 4. The operator ˜H1 acting in the space ˜H1 as

(˜H1f)(μ)=[A+2Bνi=1cosμi]f(μ)+ε1Tνf(s)ds+2ε2Tννi=1[cosμi+cossi]f(s)ds,μ=(μ1,,μn),s=(s1,,sn)Tν. (9)

Let A be an operator acting in Banach space E over C. The number λ is called regular for the operator A if the operator R(λ)=(AλI)1 , called the resolvent of the operator A, is defined throughout E and is continuous. The set of regular values of operator A is called the resolvent set of this operator, and the complement of the resolvent set is the spectrum of this operator σ(A) . The spectrum of a bounded operator is compact in C or is empty. The spectrum of a linear bounded operator is not empty. A discrete spectrum σpp(A) is a set of such λ, in which the operator AIλ is not injective.

The number λ is called the eigenvalue of the operator A, if there exists such a nonzero vector x that the equality A(x)=λx is valid. Any nonzero vector x0 , satisfying this equation is called the eigenvector of the operator A, corresponding to the eigenvalue of λ.

The discrete spectrum is all the eigenvalues of the operator A.

A continuous spectrum σcont(A) is a set of values λ, for which the resolvent (AλI)1 is not defined everywhere in a dense set in E, but is not continuous (that is, the operator AλI is injective, but not surjective, and its image is dense everywhere).

The set of all isolated, finite-fold eigenvalues of operator A is called the discrete spectrum of this operator and is denoted by σdisc(A) .

The entire spectrum of A without the discrete spectrum of this operator is called the essential spectrum of this operator A, and is denoted by σess(A) .

It is clear that the continuous spectrum of operator ˜H1 is independent of the numbers ε1 and ε2 , and is equal to segment [mν,Mν]=[A2Bν,A+2Bν] , where mν=minxTνh(x) , Mν=maxxTνh(x) (here h(x)=A+2Bνi=1cosxi ).

To find the eigenvalues and eigenfunctions of operator ˜H1 we rewrite (9) in following form:

{A+2Bνi=1cosμiz}f(μ)+ε1Tνf(s)ds+2ε2Tννi=1[cosμi+cossi]f(s)ds=0, (10)

where zR .

Suppose first that ν=1 and denote a=Tf(s)ds , b=Tf(s)cossds , h(μ)=A+2Bcos . From (10) it follows that

f(μ)=(ε1+2ε2cosμ)a+2ε2bh(μ)z. (11)

Now substitute (10) in expressing of a and b we get the following system of two linear homogeneous algebraic equations:

(1+Tε1+2ε2cossh(s)zds)a+2ε2Tdsh(s)zb=0;

Tcoss(ε1+2ε2coss)h(s)zdsa+(1+2ε2Tcossdsh(s)z)b=0.

This system has a nontrivial solution if and only if the determinant Δ1(z) of this system is equal to zero, where

Δ1(z)=(1+T(ε1+2ε2coss)dsh(s)z)(1+2ε2Tcossdsh(s)z)2ε2Tdsh(s)zTcoss(ε1+2ε2coss)h(s)zds.

Therefore, it is true the following

Lemma 3. If a real number z[m1,M1] then z is an eigenvalue of the operator ˜H1 if and only if Δ1(z)=0 .

The following Theorem describe of the exchange of the spectrum of operator ˜H1 in the case ν=1 . We consider every possible cases.

Theorem 5. Let ν=1 . Then

A).1). If ε2=B and ε1<2B , then the operator ˜H1 has a unique eigenvalue z=A+ε1 , lying the below of the continuous spectrum of the operator ˜H1 .

2). If ε2=B and ε1>2B , then the operator ˜H1 has a unique eigenvalue z=A+ε1 , lying the above of the continuous spectrum of the operator ˜H1 .

B). 1). If ε1<0 and ε2=2B or ε2=0 , then the operator ˜H1 has a unique eigenvalue z=A4B2+ε21 , lying the below of the continuous spectrum of the operator ˜H1 .

2). If ε1>0 and ε2=2B or ε2=0 , then the operator ˜H1 has a unique eigenvalue z=A+4B2+ε21 , lying the above of the continuous spectrum of the operator ˜H1 .

C). If ε1=0 and ε2>0 or ε1=0 and ε2<2B , then the operator ˜H1 has a two eigenvalues z1=A2BEE21 , and z2=A+2BEE21 , where E=(B+ε2)2ε22+2Bε2 , lying the below and above of the continuous spectrum of the operator ˜H1 .

D). 1). If ε1=2(ε22+2Bε2)B , then the operator ˜H1 has a unique eigenvalue z=A+2B(E2+1)E21 , where E=(B+ε2)2ε22+2Bε2 , lying the above of the continuous spectrum of the operator ˜H1 .

2). If ε1=2(ε22+2Bε2)B , then the operator ˜H1 has a unique eigenvalue z=A2B(E2+1)E21 , where E=(B+ε2)2ε22+2Bε2 , lying the below of the continuous spectrum of the operator ˜H1 .

E). 1). If ε2>0 and ε1>2(ε22+2Bε2)B , then the operator ˜H1 has a unique eigenvalue z=A+2B(α+EE21+α2)E21 , where E=(B+ε2)2ε22+2Bε2 , and the real number α>1 , lying the above of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and ε1>2(ε22+2Bε2)B ), then the operator ˜H1 has a unique eigenvalue z=A+2B(α+EE21+α2)E21 , where E=(B+ε2)2ε22+2Bε2 , and the real number α>1 , lying the above of the continuous spectrum of the operator ˜H1 .

F). 1). If ε2>0 and ε1<2(ε22+2Bε2)B , then the operator ˜H1 has a unique eigenvalue z=A2B(α+EE21+α2)E21<m1 , where E=(B+ε2)2ε22+2Bε2 , and the real number α>1 , lying the below of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and ε1<2(ε22+2Bε2)B , then the operator ˜H1 has a unique eigenvalue z=A2B(α+EE21+α2)E21<m1 , where E=(B+ε2)2ε22+2Bε2 , and the real number α>1 , lying the below of the continuous spectrum of the operator ˜H1 .

K). 1). If ε2>0 and 0<ε1<2(ε22+2Bε2)B , then the operator ˜H1 has a exactly two eigenvalues z1=A+2B(αEE21+α2)E21<m1 , and z2=A+2B(α+EE21+α2)E21>M1 , where E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 , lying correspondingly, the below and above of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and 0<ε1<2(ε22+2Bε2)B , then the operator ˜H1 has a exactly two eigenvalues z1=A+2B(αEE21+α2)E21<m1 , and z2=A+2B(α+EE21+α2)E21>M1 , where E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 , lying correspondingly, the below and above of the continuous spectrum of the operator ˜H1 .

M). 1). If ε2>0 and 2(ε22+2Bε2)B<ε1<0 , then the operator ˜H1 has a exactly two eigenvalues z1=A2B(α+EE21+α2)E21<m1 , and z2=A2B(αEE21+α2)E21>M1 , where E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 , lying, correspondingly the below and above of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and 2(ε22+2Bε2)B<ε1<0 , then the operator ˜H1 has a exactly two eigenvalues z1=A2B(α+EE21+α2)E21<m1 , and z2=A2B(αEE21+α2)E21>M1 , where E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 , lying, correspondingly the below and above of the continuous spectrum of the operator ˜H1 .

N). If 2B<ε2<0 , then the operator ˜H1 has no eigenvalues lying the outside of the continuous spectrum of the operator ˜H1 .

Here we will not give a proof of Theorem 5. For the proof of this theorem, see the proof of Theorem 8 from [19] , pp. 2750-2757.

Now we consider the two-dimensional case. In two-dimensional case, we have, what the equation Δ2(z)=0 , is equivalent to the equation of the form (ε2+B)2+{ε1B2+(ε22+2Bε2)(zA)}J(z)=0 , where J(z)=T2ds1ds2A+2B(coss1+coss2)z . In this case, also J(z)+0 , as z , and J(z)+ , as zm20 , and J(z) , as zM2+0 , and J(z)0 , as z+ . In one- and two-dimensional case the behavior of function J(z) be similarly. Therefore, we have the analogously results, what is find the one-dimensional case.

We consider the three-dimensional case. We denote by W Watson integral [26]

W=1π3ππππππ3dxdydz3cosxcosycosz1,516.

In the three-dimensional case, the integral T3ds1ds2ds33+coss1+coss2+coss2=T3ds1ds2ds33coss1coss2coss2 have the finite value. Expressing these integral via Watson integral W, and taking into account, what the measure is normalized, we have, that J(z)=W6B .

The following Theorem describe of the exchange of the spectrum of operator ˜H1 in the case ν=3 .

Theorem 6. Let ν=3 . Then

A). 1). If ε2=B and ε1<6B , then the operator ˜H1 has a unique eigenvalue z=A+ε1 , lying the below of the continuous spectrum of the operator ˜H1 .

2). If ε2=B and ε1>6B , then the operator ˜H1 has a unique eigenvalue z=A+ε1 , lying the above of the continuous spectrum of the operator ˜H1 .

3). If ε2=B and 6Bε1<2B , then the operator ˜H1 has no eigenvalue, lying the below of the continuous spectrum of the operator ˜H1 .

4). If ε2=B and 2B<ε16B then the operator ˜H1 has no eigenvalue, lying the above of the continuous spectrum of the operator ˜H1 .

B). 1). If ε2=2B or ε2=0 and ε1<0 , ε16BW , then the operator ˜H1 has a unique eigenvalue z1, lying the below of the continuous spectrum of the operator ˜H1 . If ε2=0 and ε1<0 , and 6BWε1<0 , then the operator ˜H1 has no eigenvalue the outside of the continuous spectrum of operator ˜H1 .

2). If ε2=2B or ε2=0 and ε1>0 , ε16BW , then the operator ˜H1 has a unique eigenvalue z2, lying the above of the continuous spectrum of the operator ˜H1 . If ε2=0 and ε1>0 , and 0<ε16BW , then the operator ˜H1 has no eigenvalue the outside of the continuous spectrum of operator ˜H1 .

C). 1). If ε1=0 and ε2>0 , E<W , then the operator ˜H1 has a unique eigenvalue z, where E=(B+ε2)2ε22+2Bε2 , lying the below of the continuous spectrum of the operator ˜H1 . If ε1=0 and ε2>0 , E>W , then the operator ˜H1 has no eigenvalues the outside the continuous spectrum of the operator ˜H1 .

2). If ε1=0 and ε2<2B , E<W , then the operator ˜H1 has a unique eigenvalue ˜z , where E=(B+ε2)2ε22+2Bε2 , lying the above of the continuous spectrum of the operator ˜H1 . If ε1=0 and ε2<2B , E>W , then the operator ˜H1 has no eigenvalues the outside the continuous spectrum of the operator ˜H1 .

D). 1). If ε1=2(ε22+2Bε2)B and E<43W , then the operator ˜H1 has a unique eigenvalue z, lying the above of the continuous spectrum of the operator ˜H1 .

2). If ε1=2(ε22+2Bε2)B and E<43W , then the operator ˜H1 has a unique eigenvalue ˜z , lying the below of the continuous spectrum of the operator ˜H1 .

E). 1). If ε2>0 and ε1>2(ε22+2Bε2)B and E<(1+α3)W , and the real number α>1 , then the operator ˜H1 has a unique eigenvalue z, lying the above of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and ε1>2(ε22+2Bε2)B and E<(1+α3)W , and the real number α>1 , then the operator ˜H1 has a unique eigenvalue z, lying the above of the continuous spectrum of the operator ˜H1 .

F). 1). If ε2>0 and ε1<2(ε22+2Bε2)B and E<(1+α3)W , and the real number α>1 , then the operator ˜H1 has a unique eigenvalue z1, lying the below of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and ε1<2(ε22+2Bε2)B and E<(1+α3)W , and the real number α>1 , then the operator ˜H1 has a unique eigenvalue z1, lying the below of the continuous spectrum of the operator ˜H1 .

K). 1). If ε2>0 and 0<ε1<2(ε22+2Bε2)B and (1α3)W<E<(1+α3)W , and the real number 0<α<1 , then the operator ˜H1 has a exactly two eigenvalues z1 and z2, lying the above and below of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and 0<ε1<2(ε22+2Bε2)B and (1α3)W<E<(1+α3)W , and the real number 0<α<1 , then the operator ˜H1 has a exactly two eigenvalues z1 and z2, lying the above and below of the continuous spectrum of the operator ˜H1 .

M). 1). If ε2>0 and 2(ε22+2Bε2)B<ε1<0 and (1α3)W<E<(1+α3)W , and the real number 0<α<1 , then the operator ˜H1 has a exactly two eigenvalues z1 and z2, lying the above and below of the continuous spectrum of the operator ˜H1 .

2). If ε2<2B and 2(ε22+2Bε2)B<ε1<0 and (1α3)W<E<(1+α3)W , and the real number 0<α<1 , then the operator ˜H1 has a exactly two eigenvalues z1 and z2, lying the above and below of the continuous spectrum of the operator ˜H1 .

N). If 2B<ε2<0 , then the operator ˜H1 has no eigenvalues lying the outside of the continuous spectrum of the operator ˜H1 .

Proof. In the case ν=3 , the continuous spectrum of the operator ˜H1 coincide with segment [m3,M3]=[A6B,A+6B] . Expressing all integrals in the equation

Δ3(z)=(1+T3(ε1+2ε23i=1cossi)ds1ds2ds3A+2B3i=1cossiz)(1+6ε2T3cossids1ds2ds3A+2B3i=1cossiz)6ε2T3ds1ds2ds3A+2B3i=1cossizT3(ε1+2ε23i=1cossi)coss1ds1ds2ds3A+2B3i=1cossiz=0

through the integral J(z)=T3ds1ds2ds3A+2B3i=1cossiz , we find that the equation Δ3(z)=0 is equivalent to the equation

[ε1B2+(ε22+2Bε2)(zA)]J(z)+(B+ε2)2=0. (12)

Moreover, the function J(z)=T3ds1ds2ds3A+2B3i=1cossiz is a differentiable function on the set \[m3,M3] , in addition, J(z)=T3ds1ds2ds3[A+2B3i=1cossiz]2>0 , z[m3,M3] .

Thus the function J(z) is an monotone increasing function on (,m3) and on (M3,+) . Furthermore, in the three-dimensional case J(z)+0 at z , and J(z)=W6B as z=A6B , and J(z)0 as z+ , and J(z)=W6B as z=A+6B .

If ε1B2+(ε22+2Bε2)(zA)0 then from (12) follows that

J(z)=(B+ε2)2ε1B2+(ε22+2Bε2)(zA).

The function ψ(z)=(B+ε2)2ε1B2+(ε22+2Bε2)(zA) has a point of asymptotic discontinuity z0=AB2ε1ε22+2Bε2 . Since ψ(z)=(B+ε2)2(ε22+2Bε2)[ε1B2+(ε22+2Bε2)(zA)]2 for all zz0 it follows that the function ψ(z) is an monotone increasing (decreasing) function on (,z0) and on (z0,+) in the case ε22+2Bε2>0 (respectively, ε22+2Bε2<0 ), in addition, and if ε2>0 , or ε2<2B , then ψ(z)+0 as z , ψ(z)+ as zz00 , ψ(z) as zz0+0 , ψ(z)0 as z+ (respectively, if 2B<ε2<0 , then ψ(z)0 as z , ψ(z) as zz00 , ψ(z)+ as zz0+0 , ψ(z)+0 as z+ ).

A). If ε2=B and ε1<6B (respectively, ε2=B and ε1>6B ), then the equation for eigenvalues and eigenfunctions (12) has the form:

{ε1B2B2(zA)}J(z)=0. (13)

It is clear, that J(z)0 for the values zσcont(˜H1) . Therefore, ε1z+A=0 , i.e., z=A+ε1 . If ε1<6B , then this eigenvalue lying the below of the continuous spectrum of operator ˜H1 , if ε1>6B , then this eigenvalue lying the above of the continuous spectrum of operator ˜H1 . If 6Bε1<2B (respectively, 2B<ε16B ), then this eigenvalue not lying in the outside of the continuous spectrum of operator ˜H1 .

B). If ε2=2B or ε2=0 and ε1<0 (respectively, ε2=2B or ε2=0 and ε1>0 ), then the equation for the eigenvalues and eigenfunctions has the form ε1B2J(z)+B2=0 , that is, J(z)=1ε1 . In the three-dimensional case J(z)+0 as z , and J(z)=W6B as z=A6B , and J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, in order to the equation J(z)=1ε1 in the below (respectively, above) of continuous spectrum of operator ˜H1 have the solution, one should implements the inequality 1ε1<W6B (respectively, 1ε1>W6B ), i.e., ε1<6BW , ε1<0 (respectively, ε1>6BW , ε1>0 ). If 6BW<ε1<0 (respectively, 0<ε1<6BW ), then the operator ˜H1 has no eigenvalues the outside the continuous spectrum of operator ˜H1 .

C). If ε1=0 and ε2>0 (respectively, ε1=0 and ε2<2B ), then the equation for the eigenvalues and eigenfunctions take in the form

(ε22+2Bε2)(zA)J(z)=(B+ε2)2,

or

J(z)=(B+ε2)2(ε22+2Bε2)(zA).

Denote E=(B+ε2)2ε22+2Bε2 . Then J(z)=EzA , or J(z)=EAz . In the three-dimensional case J(z)+0 as z , and J(z)=W6B as z=A6B , and J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, in order to the equation J(z)=EzA in the below (respectively, above) of continuous spectrum of operator ˜H1 have the solution, one should implements the inequality E6B<W6B (respectively, E6B>W6B ), i.e., E<W . If ε1=0 and ε2>0 , E>W (respectively, ε1=0 and ε2<2B , E>W ), then the operator ˜H1 has no eigenvalues the outside the continuous spectrum of operator ˜H1 .

D). If ε1=2(ε22+2Bε2)B , then the equation for eigenvalues and eigenfunctions has the form

(ε22+2Bε2)(zA+2B)J(z)=(B+ε2)2,

from this we have equation in the form:

J(z)=(B+ε2)2(ε22+2Bε2)(zA+2B). (14)

We denote E=(B+ε2)2ε22+2Bε2 . In the first we consider Equation (14) in the below of continuous spectrum of operator ˜H1 . In the below of continuous spectrum of operator ˜H1 , the function EAz2B+0 , as z , EAz2B=E4B , as z=A6B , and in the three-dimensional case J(z)+0 as z , and J(z)=W6B as z=A6B , and J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, the below of continuous spectrum of operator ˜H1 , the equation J(z)=EAz2B has a unique solution, if E4B>W6B , i.e., E>2W3 . This inequality incorrectly. Therefore, the below of continuous spectrum of operator ˜H1 , this equation has no solution.

We now consider the equation for eigenvalues and eigenfunctions J(z)=EzA+2B , in the above of continuous spectrum of operator ˜H1 . In the above of continuous spectrum of operator ˜H1 , the function EAz2B0 , as z+ , EAz2B=E8B , as z=A+6B , and in the three-dimensional case J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, the above of continuous spectrum of operator ˜H1 , the equation J(z)=EAz2B has a unique solution, if E8B>W6B , i.e., E<4W3 . This inequality correctly. Therefore, the above of continuous spectrum of operator ˜H1 , this equation has a unique solution z.

If ε1=2(ε22+2Bε2)B , then the equation for eigenvalues and eigenfunctions has the form

(ε22+2Bε2)(zA2B)J(z)=(B+ε2)2,

from this we have the equation in the form (14).

We denote E=(B+ε2)2ε22+2Bε2 . In the first we consider the equation (14) in the below of continuous spectrum of operator ˜H1 . In the below of continuous spectrum of operator ˜H1 , the function EAz+2B+0 , as z , EAz+2B=E8B , as z=A6B , and in the three-dimensional case J(z)+0 as z , and J(z)=W6B as z=A6B , and J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, the below of continuous spectrum of operator ˜H1 , the equation J(z)=EAz+2B has a unique solution, if E8B<W6B , i.e., E<4W3 . This inequality correctly. Therefore, the below of continuous spectrum of operator ˜H1 , this equation has a unique solution.

We now consider the equation for eigenvalues and eigenfunctions J(z)=EzA2B , in the above of continuous spectrum of operator ˜H1 . In the above of continuous spectrum of operator ˜H1 , the function EAz+2B0 , as z+ , EAz+2B=E4B , as z=A+6B , and in the three-dimensional case J(z)0 as z+ , and J(z)=W6B as z=A+6B . Therefore, the above of continuous spectrum of operator ˜H1 , the equation J(z)=EAz+2B has a unique solution, if E4B>W6B , i.e., E<2W3 . This inequality incorrectly. Therefore, the above of continuous spectrum of operator ˜H1 , this equation has no solution.

E). If ε2>0 and ε1>2(ε22+2Bε2)B , (respectively, ε2<2B and ε1>2(ε22+2Bε2)B ), then consider necessary, that ε1=α×2(ε22+2Bε2)B , where α>1 real number. Then the equation for eigenvalues and eigenfunctions has the form {α×2(ε22+2Bε2)B×B2+(ε22+2Bε2)(zA)}J(z)+(B+ε2)2=0 , or (ε22+2Bε2)(zA+2αB)J(z)+(B+ε2)2=0 . From this J(z)=(B+ε2)2(ε22+2Bε2)(zA+2αB) . We denote E=(B+ε2)2ε22+2Bε2 , then J(z)=EzA+2αB . In the first we consider this equation in the below of the continuous spectrum of operator ˜H1 . Then J(z)+0 , as z , J(z)=W6B , as z=A6B , EzA+2αB+0 , as z , and EzA+2αB=E(62α)B , as z=A6B . The equation J(z)=EzA+2αB have a unique solution, if E(62α)B<W6B . From here E<(3α)W3 . This inequality is incorrect. Therefore, the below of continuous spectrum of operator ˜H1 , the operator ˜H1 has no eigenvalues.

The above of continuous spectrum of operator ˜H1 , we have the J(z)0 , if z+ , J(z)=W6B , if z=A6B . Besides, EzA+2αB0 , as z+ , EzA+2αB=E6B+2αB , if z=A+6B .

The equation J(z)=EzA+2αB have a unique solution, if E(6+2α)B>W6B . From here E<(3+α)W3 . This inequality is correctly. Therefore, the above of continuous spectrum of operator ˜H1 , the operator ˜H1 has a unique eigenvalues z1.

F). If ε2>0 and ε1<2(ε22+2Bε2)B (respectively, ε2<2B and ε1<2(ε22+2Bε2)B ), then we assume that ε1=α×2(ε22+2Bε2)B , where α>1 real number. The equation for eigenvalues and eigenfunctions take in the form

(ε22+2Bε2)(zA2αB)J(z)=(B+ε2)2.

From here

J(z)=(B+ε2)2(ε22+2Bε2)(zA2αB).

The introduce notation E=(B+ε2)2ε22+2Bε2 . Then we have the equation in the form:

J(z)=EzA2αB. (15)

In the below of the continuous spectrum of operator ˜H1 , we have the equation J(z)=EAz+2αB . In the below of continuous spectrum of operator ˜H1 , EzA2αB+0 , as z , EzA2αB=E6B+2αB , as z=A6B .

The equation J(z)=EzA+2αB have a unique solution, if E(6+2α)B<W6B . From here E<(3+α)W3 . This inequality is correctly. Therefore, the below of continuous spectrum of operator ˜H1 , the operator ˜H1 has a unique eigenvalues.

In the above of continuous spectrum of operator ˜H1 , EzA2αB0 , as z , EzA2αB=E6B2αB , as z=A+6B . Therefore, the above of continuous spectrum of operator ˜H1 , the operator ˜H1 has a unique eigenvalues, if E6B2αB>W6B . From here E<(3α)W3 , what is incorrectly. Therefore, the above of continuous spectrum of operator ˜H1 , the operator ˜H1 has no eigenvalues.

K). If ε2>0 and 0<ε1<2(ε22+2Bε2)B (respectively, ε2<2B and 0<ε1<2(ε22+2Bε2)B ), the we take ε1=α×2(ε22+2Bε2)B , where 0<α<1 positive real number. Then the equation for eigenvalues and eigenfunctions has the form:

(ε22+2Bε2)(zA+2αB)J(z)=(B+ε2)2,0<α<1. (16)

We denote E=(B+ε2)2ε22+2Bε2 . Then the Equation (16) receive the form

J(z)=EzA+2αB.

In the below of continuous spectrum of operator ˜H1 , we have EzA+2αB+0 , as z , and EzA+2αB=E2B(3α) , as z=A6B . The equation J(z)=EzA+2αB have a unique solution the below of continuous spectrum of operator ˜H1 , if E(62α)B>W6B . From here E>(3α)W3 . This inequality is correctly. Therefore, the below of continuous spectrum of operator ˜H1 , the operator ˜H1 has a unique eigenvalues z1.

The above of continuous spectrum of operator ˜H1 , we have EzA+2αB0 , as z+ , and EzA+2αB=E2B(3+α) , as z=A+6B . The equation J(z)=EzA+2αB have a unique solution the above of operator ˜H1 , if E2B(3+α)>W6B , i.e., E<(3+α)W3 . This inequality is correctly.

Consequently, in this case the operator ˜H1 have two eigenvalues z1 and z2, lying the below and above of continuous spectrum of operator ˜H1 .

M). If ε2>0 and 2(ε22+2Bε2)B<ε1<0 (respectively, ε2<2B and 2(ε22+2Bε2)B<ε1<0 ), the we take ε1=α×2(ε22+2Bε2)B , where 0<α<1 positive real number. Then the equation for eigenvalues and eigenfunctions has the form:

(ε22+2Bε2)(zA2αB)J(z)=(B+ε2)2,0<α<1. (17)

We denote E=(B+ε2)2ε22+2Bε2 . Then the Equation (17) receive the form

J(z)=EzA2αB.

In the below of continuous spectrum of operator ˜H1 , we have EzA2αB+0 , as z , and EzA2αB=E2B(3+α) , as z=A6B . The equation J(z)=EzA2αB have a unique solution the below of continuous spectrum of operator ˜H1 , if E(6+2α)B<W6B . From here E<(3+α)W3 . This inequality is correctly. Therefore, the below of continuous spectrum of operator ˜H1 , the operator ˜H1 has a unique eigenvalues z1.

The above of continuous spectrum of operator ˜H1 , we have EzA2αB0 , as z+ , and EzA2αB=E2B(3α) , as z=A+6B . The equation J(z)=EzA2αB have a unique solution the above of continuous spectrum of operator ˜H1 , if E2B(3α)<W6B , i.e., E>(3α)W3 . This inequality is correctly.

Consequently, in this case the operator ˜H1 have two eigenvalues z1 and z2, lying the below and above of ˜H1 .

N). If 2B<ε2<0 , then ε22+2Bε2<0 , and the function ψ(z)=(B+ε2)2ε1B+(ε22+2Bε2)(zA) is a decreasing function in the intervals (,z0) and (z0,+) ; By, z the function ψ(z)0 , and by zz00 , the function ψ(z) , and by z+ , ψ(z)+0 , and by zz0+0 , ψ(z)+ . The function J(z)+0 , by z , and by z=A6B , the function J(z)=W6B , and by z=A+6B , the function J(z)=W6B , by z+ , the function J(z)0 . Therefore, the equation ψ(z)=J(z) , that’s impossible the solutions in the outside the continuous spectrum of operator ˜H1 . Therefore, in this case, the operator ˜H1 has no eigenvalues lying the outside of the continuous spectrum of the operator ˜H1 .

From obtaining results is obviously, that the spectrum of operator ˜H1 is consists from continuous spectrum and no more than two eigenvalues.

Taking into account that the function f(λ,μ,γ,θ) is antisymmetric, and using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [27] , we can verify that the operator 2˜H1t can be represented in the form

˜H21tψ21t={˜H1I+I˜H1+K(λ,μ)}II+II{˜H1I+I˜H1} (18)

where

(˜H1f)(λ)={A+2Bνi=1cosλi}f(λ)+(A0A)Tνf(s)ds+2BTννi=1[cosλi+cossi]f(s)ds,

and K(λ,μ)=UTνf(s,λ+μs)ds+(U0U)TνTνf(s,k)dsdk , and I is the unit operator in the space ˜H1 .

The spectrum of the operator AI+IB , where A and B are densely defined bounded linear operators, was studied in [28] [29] [30] . Explicit formulas were given there that express the essential spectrum σess(AI+IB) and discrete spectrum σdisc(AI+IB) of operator AI+IB in terms of the spectrum σ(A) and the discrete spectrum σdisc(A) of A and in terms of the spectrum σ(B) and the discrete spectrum σdisc(B) of B:

σdisc(AI+IB)={σ(A)\σess(A)+σ(B)\σess(B)}\{(σess(A)+σ(B))(σ(A)+σess(B))}, (19)

σess(AI+IB)=(σess(A)+σ(B))(σ(A)+σess(B)). (20)

It is clear that σ(AI+IB)={λ+μ:λσ(A),μσ(B)} .

Consequently, we must investigate in first the spectrum of the operators ˜H1 .

4. Structure of the Essential Spectrum and Discrete Spectrum of Operator ˜H21t

Consequently, the operator represented of the form

˜H21t={˜Ht2+K(λ,μ)}II+II˜Ht2, (21)

where ˜Ht2=˜H1I+I˜H1 are the energy operator of two-electron systems in the impurity Hubbard model in triplet state.

We now, using the obtained results and representation (18) and (21), we first describe the structure of essential spectrum and discrete spectrum of the operator ˜Hs2=˜Ht2+K(λ,μ) .

From the beginning, we consider the operator ˜H(U)=˜Ht2+K1 .

Since, the family of the operators ˜H(U) is the family of bounded operators, that the ˜H(U) is the family of bounded operator valued analytical functions.

Therefore, in these family, one can the apply the Kato-Rellix theorem.

Theorem 7. (Kato-Rellix theorem)) [27] .

Let T(β) is the analytical family in the terms of Kato. Let E0 is a nondegenerate eigenvalue of T(β0) . Then as β , near to β0 , the exist exactly one point E(β)σ(T(β)) the near E0 and this point is isolated and nondegenerated. E(β) is an analytical function of β as β , the near to β0 , and exist the analytical eigenvector Ω(β) as β the near to β0 . If the as real ββ0 the operator T(β) is a self-adjoint operator, then Ω(β) can selected thus, that it will be normalized of real ββ0 .

Since, the operator ˜Ht2 has a nondegenerate eigenvalue, such as, the near of eigenvalue 2z1 of the operator ˜Ht2 , the operator ˜H(U) as U, near U0=0 , has a exactly one eigenvalue E(U)σ(˜H(U)) the near 2z1 and this point is isolated and nondegenerated. The E(U) is a analytical function of U as U, the near to U0=0 .

As the large values the existence no more one additional eigenvalue of the operator ˜H(U) is following from the same, what the perturbation (K1˜f)(λ,μ)=UTνf(s,λ+μs)ds is the one-dimensional operator.

A new we consider the family of operators ˜H(ε3)=˜H(U)+K2 .

As, the operator ˜H(U) has a nondegenerate eigenvalue, consequently, the near of eigenvalue E(U) the operator ˜H(U) , operator ˜H(ε3) as ε3 , the near of ε3=0 , has a exactly one eigenvalue E(ε3)σ(˜H(ε3)) the near E(U) and this point is the isolated and nondegenerated. The E(ε3) is a analytical function of ε3 , as ε3 , the near to ε3=0 .

Later on via z3, and z4 we denote the additional eigenvalues of operator ˜Hs2 . Thus, we prove the next theorems, the described the spectra of operator ˜Hs2 .

Now, using the obtained results (Theorem 5 and 6) and representation (18), and (21), we describe the structure of the essential spectrum and discrete spectrum of the operator ˜H21t .

Theorem 8. Let ν=1 . Then

A). If ε2=B and ε1<2B , or if ε2=B and ε1>2B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and the discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A+ε1 , and z3 and z4 are the additional eigenvalues of the operator ˜Hs2 .

B). 1). If ε2=2B or ε2=0 and ε1<0 , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A4B2+ε21 .

2). If ε2=2B or ε2=0 and ε1>0 , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A+4B2+ε21 .

C). If ε1=0 and ε2>0 or ε1=0 and ε2<2B , then the essential spectrum of the operator ˜H21t is consists of the union of sixteen segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z1,3A+6B+z1][3A6B+z2,3A+6B+z2][2A4B+2z1,2A+4B+z1][2A4B+2z2,2A+4B+z2][2A4B+z1+z2,2A+4B+z1+z2][A2B+3z1,A+2B+3z1][A2B+3z2,A+2B+3z2][A2B+z1+2z2,A+2B+z1+2z2][A2B+2z1+z2,A+2B+2z1+z2][2A4B+z3,2A+4B+z3][2A4B+z4,2A+4B+z4][A2B+z1+z3,2A+4B+z1+z3][A2B+z1+z4,2A+4B+z1+z4][A2B+z2+z3,2A+4B+z2+z3][A2B+z2+z4,2A+4B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of eleven eigenvalues: σdisc(˜H21t)={4z1,3z1+z2,4z2,2z1+2z2,z1+3z2,2z1+z3,z1+z2+z3,2z2+z3,2z1+z4,z1+z2+z4,2z2+z4} , where z1=A2BEE21 , and z2=A+2BEE21 , and E=(B+ε2)2ε22+2Bε2 .

D). 1). If ε1=2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A+2B(E2+1)E21 , and E=(B+ε2)2ε22+2Bε2 .

2). If ε1=2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A2B(E2+1)E21 , and E=(B+ε2)2ε22+2Bε2 .

E). If ε2>0 and ε1>2(ε22+2Bε2)B , or if ε2<2B and ε1>2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A+2B(α+EE21+α2)E21 and E=(B+ε2)2ε22+2Bε2 , and the real number α>1 .

F). If ε2>0 and ε1<2(ε22+2Bε2)B , or if ε2<2B and ε1<2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z,3A+6B+z][2A4B+2z,2A+4B+2z][A2B+3z,A+2B+3z][2A4B+z3,2A+4B+z3][A2B+z+z3,A+2B+z+z3][2A4B+z4,2A+4B+z4][A2B+z+z4,A+2B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A2B(α+EE21+α2)E21 and E=(B+ε2)2ε22+2Bε2 , and the real number α>1 .

K). If ε2>0 and 0<ε1<2(ε22+2Bε2)B , or if ε2<2B and 0<ε1<2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of sixteen segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z1,3A+6B+z1][3A6B+z2,3A+6B+z2][2A4B+2z1,2A+4B+2z1][2A4B+2z2,2A+4B+2z2][2A4B+z1+z2,2A+4B+z1+z2][A2B+3z1,A+2B+3z1][A2B+3z2,A+2B+3z2][A2B+2z1+z2,A+2B+2z1+z2][A2B+z1+2z2,A+2B+z1+2z2][2A4B+z3,2A+4B+z3][2A4B+z4,2A+4B+z4][A2B+z1+z3,A+2B+z1+z3][A2B+z1+z4,A+2B+z1+z4][A2B+z2+z3,A+2B+z2+z3][A2B+z2+z4,A+2B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of eleven eigenvalues: σdisc(˜H21t)={4z1,3z1+z2,2z1+2z2,z1+3z2,4z2,2z1+z3,z1+z2+z3,2z2+z3,2z1+z4,2z2+z4,z1+z2+z4} , where z1=A+2B(α+EE21+α2)E21 and z2=A+2B(αEE21+α2)E21 , and E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 .

M). If ε2>0 and 2(ε22+2Bε2)B<ε1<0 , or if ε2<2B and 2(ε22+2Bε2)B<ε1<0 , then the essential spectrum of the operator ˜H21t is consists of the union of sixteen segments: σess(˜H21t)=[4A8B,4A+8B][3A6B+z1,3A+6B+z1][3A6B+z2,3A+6B+z2][2A4B+2z1,2A+4B+2z1][2A4B+2z2,2A+4B+2z2][2A4B+z1+z2,2A+4B+z1+z2][A2B+3z1,A+2B+3z1][A2B+3z2,A+2B+3z2][A2B+2z1+z2,A+2B+2z1+z2][A2B+z1+2z2,A+2B+z1+2z2][2A4B+z3,2A+4B+z3][2A4B+z4,2A+4B+z4][A2B+z1+z3,A+2B+z1+z3][A2B+z1+z4,A+2B+z1+z4][A2B+z2+z3,A+2B+z2+z3][A2B+z2+z4,A+2B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of eleven eigenvalues: σdisc(˜H21t)={4z1,3z1+z2,2z1+2z2,z1+3z2,4z2,2z1+z3,z1+z2+z3,2z2+z3,2z1+z4,2z2+z4,z1+z2+z4} , where z1=A+2B(α+EE21+α2)E21 and z2=A+2B(αEE21+α2)E21 , and E=(B+ε2)2ε22+2Bε2 , and the real number 0<α<1 .

N). If 2B<ε2<0 , then the essential spectrum of the operator ˜H21t is consists of the union of three segments: σess(˜H21t)=[4A8B,4A+8B][2A4B+z3,2A+4B+z3][2A4B+z4,2A+4B+z4] , and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

Proof. A). From the representation (18), (21) and the formulas (19) and (20), and the Theorem 5, follow the in one-dimensional case, the continuous spectrum of the operator ˜H1 is consists σcont(˜H1)=[A2B,A+2B] , and the discrete spectrum of the operator ˜H1 is consists of unique eigenvalue z=A+ε1 . The operator K is a two-dimensional operator. Therefore, the essential spectrum of the operators ˜H1I+I˜H1 and ˜Hs2 coincide (see. chapter XIII, paragraph 4, in [22] ) and is consists from segments [2A4B,2A+4B] , and [A2B+z,A+2B+z] . Of extension the two-dimensional operator K to the operator ˜H1I+I˜H1 can appear no more then two additional eigenvalues z3 and z4. These give the statement A) of the Theorem 8.

B). In this case the operator ˜H1 has a one eigenvalue z1, lying the outside of the continuous spectrum of operator ˜H1 . Therefore, the essential spectrum of the operators ˜H1I+I˜H1 is consists of the union of two segments and discrete spectrum of the operator ˜H1I+I˜H1 is consists of single point. These give the statement B) of the Theorem 8. The other statements of the Theorem 8 the analogously is proved.

The next theorems is described the structure of essential spectrum of the operator ˜H21t in the three-dimensional case.

Theorem 9. Let ν=3 . Then

A).1). If ε2=B and ε1<6B , or if ε2=B and ε1>6B , then the essential spectrum of the operator ˜H21t is consists of the union of eight segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z,3A+18B+z][2A12B+2z,2A+12B+2z][A6B+3z,A+6B+3z][2A12B+z3,2A+12B+z3][A6B+z+z3,A+6B+z+z3][2A12B+z4,2A+12B+z4][A6B+z+z4,A+6B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z=A+ε1 , z3 and z4 are the additional eigenvalues of the operator ˜Hs2 .

2). If ε2=B and 6Bε1<2B , or if ε2=B and 2B<ε16B , then the essential spectrum of the operator ˜H21t is consists of the union of three segments: σess(˜H21t)=[4A24B,4A+24B][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4] , and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

B). 1). If ε2=2B or ε2=0 and ε1<0 , ε16BW , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z1,3A+18B+z1][2A12B+2z1,2A+12B+2z1][A6B+3z1,A+6B+3z1][2A12B+z3,2A+12B+z3][A6B+z1+z3,A+6B+z1+z3][2A12B+z4,2A+12B+z4][A6B+z1+z4,A+6B+z1+z4] and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z1,2z1+z3,2z1+z4} , where z1 are the eigenvalue of operator ˜H1 .

If 6BWε1<0 , then the essential spectrum of the operator ˜H21t is consists of the union of three segments: σess(˜H21t)=[4A24B,4A+24B][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4] , and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

2). If ε2=2B or ε2=0 and ε1>0 , ε16BW , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z2,3A+18B+z2][2A12B+2z2,2A+12B+2z2][A6B+3z2,A+6B+3z2][2A12B+z3,2A+12B+z3][A6B+z2+z3,A+6B+z2+z3][2A12B+z4,2A+12B+z4][A6B+z2+z4,A+6B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z2,2z2+z3,2z2+z4} , where z2 are the eigenvalue of operator ˜H1 .

If 0<ε16BW , then the essential spectrum of the operator ˜H21t is consists of the union of three segments: σess(˜H21t)=[4A24B,4A+24B][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4] , and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

C). 1). If ε1=0 and ε2>0 , E<W , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z,3A+18B+z][2A12B+2z,2A+12B+2z][A6B+3z,A+6B+3z][2A12B+z3,2A+12B+z3][A6B+z+z3,A+6B+z+z3][2A12B+z4,2A+12B+z4][A6B+z+z4,A+6B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} where z ˜z is the eigenvalue of operator ˜H1 , and E=(B+ε2)2ε22+2Bε2 . If ε1=0 and ε2>0 , E>W , then the essential spectrum of the operator ˜H21t is consists of a union of three segment: σess(˜H21t)=[4A24B,4A+24B][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4] and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

2). If ε1=0 and ε2<2B , E<W , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+˜z,3A+18B+˜z][2A12B+2˜z,2A+12B+2˜z][A6B+3˜z,A+6B+3˜z][2A12B+z3,2A+12B+z3][A6B+˜z+z3,A+6B+˜z+z3][2A12B+z4,2A+12B+z4][A6B+˜z+z4,A+6B+˜z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4˜z,2˜z+z3,2˜z+z4} , where ˜z is the eigenvalue of operator ˜H1 , and E=(B+ε2)2ε22+2Bε2 . If ε1=0 and ε2<2B , E>W , then the essential spectrum of the operator ˜H21t is consists of a union of three segment: σess(˜H21t)=[4A24B,4A+24B][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4] and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

D). 1). If ε1=2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z,3A+18B+z][2A12B+2z,2A+12B+2z][A6B+3z,A+6B+3z][2A12B+z3,2A+12B+z3][A6B+z+z3,A+6B+z+z3][2A12B+z4,2A+12B+z4][A6B+z+z4,A+6B+z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z,2z+z3,2z+z4} , where z is the eigenvalue of operator ˜H1 .

2). If ε1=2(ε22+2Bε2)B , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+˜z,3A+18B+˜z][2A12B+2˜z,2A+12B+2˜z][A6B+3˜z,A+6B+3˜z][2A12B+z3,2A+12B+z3][A6B+˜z+z3,A+6B+˜z+z3][2A12B+z4,2A+12B+z4][A6B+˜z+z4,A+6B+˜z+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4˜z,2˜z+z3,2˜z+z4} , where ˜z is the eigenvalue of operator ˜H1 .

E). If ε2>0 and ε1>2(ε22+2Bε2)B and E<(1+α3)W , or if ε2<2B and ε1>2(ε22+2Bε2)B and E<(1+α3)W , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z1,3A+18B+z1][2A12B+2z1,2A+12B+2z1][A6B+3z1,A+6B+3z1][2A12B+z3,2A+12B+z3][A6B+z1+z3,A+6B+z1+z3][2A12B+z4,2A+12B+z4][A6B+z1+z4,A+6B+z1+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z1,2z1+z3,2z1+z4} , where z1 is the eigenvalue of operator ˜H1 .

F). If ε2>0 and ε1<2(ε22+2Bε2)B and E<(1+α3)W , or if ε2<2B and ε1<2(ε22+2Bε2)B and E<(1+α3)W , then the essential spectrum of the operator ˜H21t is consists of the union of eighth segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z1,3A+18B+z1][2A12B+2z1,2A+12B+2z1][A6B+3z1,A+6B+3z1][2A12B+z3,2A+12B+z3][A6B+z1+z3,A+6B+z1+z3][2A12B+z4,2A+12B+z4][A6B+z1+z4,A+6B+z1+z4] , and discrete spectrum of the operator ˜H21t is consists of three eigenvalues: σdisc(˜H21t)={4z1,2z1+z3,2z1+z4} , where z1 is the eigenvalue of operator ˜H1 .

K). If ε2>0 and 0<ε1<2(ε22+2Bε2)B and E<(1α3)W , or if ε2<2B and 0<ε1<2(ε22+2Bε2)B and E<(1α3)W , then the essential spectrum of the operator ˜H21t is consists of the union of sixteen segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z1,3A+18B+z1][3A18B+z2,3A+18B+z2][2A12B+2z1,2A+12B+2z1][2A12B+z1+z2,2A+12B+z1+z2][2A12B+2z2,2A+12B+2z2][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4][A6B+3z1,A+6B+3z1][A6B+2z1+z2,A+6B+2z1+z2][A6B+z1+2z2,A+6B+z1+2z2][A6B+3z2,A+6B+3z2][A6B+z1+z3,A+6B+z1+z3][A6B+z1+z4,A+6B+z1+z4][A6B+z2+z3,A+6B+z2+z3][A6B+z2+z4,A+6B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of eleven eigenvalues: σdisc(˜H21t)={4z1,3z1+z2,2z1+2z2,2z1+z3,2z1+z4,z1+3z2,z1+z2+z3,z1+z2+z4,4z2,2z2+z3,2z2+z4} , where z1 and z2 are the eigenvalues of operator ˜H1 .

M). If ε2>0 and 2(ε22+2Bε2)B<ε1<0 and E<(1+α3)W , or if ε2<2B and 2(ε22+2Bε2)B<ε1<0 and E<(1+α3)W , then the essential spectrum of the operator ˜H21t is consists of the union of sixteen segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z1,3A+18B+z1][3A18B+z2,3A+18B+z2][2A12B+2z1,2A+12B+2z1][2A12B+z1+z2,2A+12B+z1+z2][2A12B+2z2,2A+12B+2z2][2A12B+z3,2A+12B+z3][2A12B+z4,2A+12B+z4][A6B+3z1,A+6B+3z1][A6B+2z1+z2,A+6B+2z1+z2][A6B+z1+2z2,A+6B+z1+2z2][A6B+3z2,A+6B+3z2][A6B+z1+z3,A+6B+z1+z3][A6B+z1+z4,A+6B+z1+z4][A6B+z2+z3,A+6B+z2+z3][A6B+z2+z4,A+6B+z2+z4] , and discrete spectrum of the operator ˜H21t is consists of eleven eigenvalues: σdisc(˜H21t)={4z1,3z1+z2,2z1+2z2,2z1+z3,2z1+z4,z1+3z2,z1+z2+z3,z1+z2+z4,4z2,2z2+z3,2z2+z4} , where z1 and z2 are the eigenvalues of operator ˜H1 .

N). If 2B<ε2<0 , then the essential spectrum of the operator ˜H21t is consists of a union of three segments: σess(˜H21t)=[4A24B,4A+24B][3A18B+z3,3A+18B+z3][3A18B+z4,3A+18B+z4] , and discrete spectrum of the operator ˜H21t is consists of empty set: σdisc(˜H21t)= .

Proof. A). 1). From the Theorem 6 is follows, that, if ν=3 and ε2=B and ε1<6B (respectively, ε2=B and ε1>6B ), the operator ˜H1 has a unique eigenvalue z=A+ε1 , the outside the continuous spectrum of the operator ˜H1 . Furthermore, the continuous spectrum of the operator ˜H1 is consists of the segment [A6B,A+6B] , therefore, the essential spectrum of the operator ˜Hs2 is consists of a union of two segments: σess(˜Hs2)=[2A12B,2A+12B][A6B+z,A+6B+z] . The number 2z is the eigenvalue for the operator ˜Hs2 . In the representation (18) and (21) the operator K is a two-dimensional operator. Therefore, the operator ˜Hs2 can have two additional eigenvalues z3 and z4 . Consequently, the operator ˜Hs2 can have no more than three eigenvalues 2z,z3 and z4 .

2). From the Theorem 6 is follows, that, if ν=3 and ε2=B and 6Bε1<2B (respectively, ε2=B and 2B<ε16B ), then the operator ˜H1 has no eigenvalues, the outside the continuous spectrum of the operator ˜H1 . Furthermore, the continuous spectrum of the operator ˜H1 is consists of the segment [A6B,A+6B] , therefore, the essential spectrum of the operator ˜Hs2 is consists of a single segment: σess(˜Hs2)=[2A12B,2A+12B] . In the representation (18) and (21) the operator K is a two-dimensional operator. Therefore, the operator ˜Hs2 can have two additional eigenvalues z3 and z4 . Consequently, the operator ˜Hs2 can have no more than two eigenvalues z3 and z4 .

M). From the Theorem 6 is follows, that, if ν=3 and ε2>0 and 2(ε22+2Bε2)B<ε1<0 and E<(1+α3)W (respectively, ε2<2B and 2(ε22+2Bε2)B<ε1<0 and E<(1+α3)W ), the operator ˜H1 has a exactly two eigenvalues z1 and z2 , lying the below and above of the continuous spectrum of the operator ˜H1 . Furthermore, the continuous spectrum of the operator ˜H1 is consists of the segment [A6B,A+6B] , therefore, then the essential spectrum of the operator ˜Hs2 is consists of the union of three segments: σess(˜Hs2)=[2A12B,2A+12B][A6B+z1,A+6B+z1][A6B+z2,A+6B+z2] , and point 2z1,2z2 and z1+z2 , are the eigenvalues of the operator ˜H1I+I˜H1 , and in the representation (18) and (21) the operator K is a two-dimensional operator. Therefore, the operator ˜Hs2 can have two additional eigenvalues z3 and z4 . Consequently, the operator ˜Hs2 can have no more than five eigenvalues 2z1,z1+z2,2z2,z3 and z4 .

The other statements of the Theorem 9 the analogously is proved.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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