STRUCTURE OF ESSENTIAL SPECTRA AND DISCRETE SPECTRUM OF THE ENERGY OPERATOR OF FOUR-ELECTRON SYSTEMS IN THE IMPURITY HUBBARD MODEL. QUINTET STATE. ONE-DIMENSIONAL CASE

Tashpulatov S. M.; Parmanova R. T.

XAK: 517.984

MSC2010: 46L60, 47L90, 70H06, 70F05.

STRUCTURE OF ESSENTIAL SPECTRA AND DISCRETE SPECTRUM OF THE ENERGY OPERATOR OF FOUR-ELECTRON SYSTEMS IN THE IMPURITY HUBBARD MODEL. QUINTET STATE. ONE-DIMENSIONAL CASE

Institute of Nuclear Physics of Academy of Science of Republic of Uzbekistan 100214, Republic of Uzbekistan, Tashkent, M. Ulugbek region, M. Ulugbek village, U. Gulyamov st. 1. e-mail: sadullatashpulatov@yandex.ru, toshpul@mail.ru, togaymurodota@gmail.com

Structure of essential spectra and discrete spectrum of the energy operator of four-electron systems in the impurity Hubbard model. Quintet state. One-dimensional case.

Tashpulatov S. M. and Parmanova R. T.

Abstract. We consider the energy operator of four electron systems in the impurity Hubbard model and investigated the structure of essential spectra and discrete spectrum of the system in the quintet state of the system. It is shown that there are such situations: a). the essential spectrum of the four-electron quintet state operator is consists of the union of four segments, and the discrete spectrum of the four-electron quintet state operator is consists of single eigenvalue; b). the essential spectrum of the four-electron quintet state operator is consists of the union of ten segments, and the discrete spectrum of the four-electron quintet state operator is consists of five eigenvalues; c). the essential spectrum of the four-electron quintet state operator is consists of single segment, and the discrete spectrum of the four-electron quintet state operator is empty set; Provided that every situation occurs. Found the conditions, when every situation to take place.

Keywords: Impurity Hubbard model, four-electron system, essential spectra, discrete spectrum, quintet state, triplet state, singlet state

Introduction

In the early 1970s, three papers [2, 3, 5], where a simple model of a metal was proposed that has become a fundamental model in the theory of strongly correlated electron systems, appeared almost simultaneously and independently. In that model, a single nondegenerate electron band with a local Coulomb interaction is considered. The model Hamiltonian contains only two parameters: the matrix element t of electron hopping from a lattice site to a neighboring site and the parameter U of the on-site Coulomb

repulsion of two electrons. In the secondary quantization representation, the Hamiltonian can be written as

H = am,Y+ ain,tam,tain,lam,i, (1)

m, Y m

where aOmY and am,Y denote Fermi operators of creation and annihilation of an electron with spin y on a site m and the summation over t means summation over the nearest neighbors on the lattice.

The model proposed in [2, 3, 5] 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 [1]. We also recall that the Hubbard model is a particular case of the Shubin-Wonsowsky polaron model [11], which had appeared 30 years before [2, 3, 5]. 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 simplicity and sufficiency of Hamiltonian (1) have made the Hubbard model very popular and effective for describing strongly correlated electron systems.

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 longrange 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 orbitals 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 [4, 6-8, 12]. Therefore, obtaining exact results for the spectrum and wave functions of the crystal described by the 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 [6]. It is known that two-electron systems can be in two states, triplet and singlet [4, 6-8, 12]. It was proved in [6] that the spectrum of the system Hamiltonian H in the triplet state is purely continuous and coincides with a segment [m, M] = [2A — 4Bv, 2A + 4Bv], 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 U — 0 and increases without bound as U —y to. 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 [13]. In the three-electron systems are exists quartet state, and two type doublet states. The quartet state corresponds to the free motion of three electrons over the lattice with the basic functions g^n ,p = a^ t®+ t®+t In the work [13] is proved that the essential spectrum of the system in a quartet state consists of a single segment and the three-electron bound state or the three-electron antibound state is absent. The doublet state corresponds to the basic functions

1<dm%,p = am,ta+,4a3+t^o, and 2di1l/2n,p = am,tap,tap4"q. If v =1 and U > 0, then the essential spectrum of the system of first doublet state operator Hf is exactly the union of three segments and the discrete spectrum of Hf consists of a single point, i.e., in the system exists unique antibound state. In the two-dimensional case, we have the analogous results. In the three-dimensional case, or the essential spectrum of the system in the first doublet state operator Hf is the union of three segments and the discrete spectrum of operator Hf consists of a single point, i.e., in the system exists only one antibound state, or the essential spectrum of the system in the first doublet state operator Hf is the union of two segments and the discrete spectrum of the operator Hf is empty, or the essential spectrum of the system in the first doublet state operator Hf is consists of a single segment, and discrete spectrum is empty, i.e., in the system the antibound state is absent. In the one-dimensional case, the essential spectrum of the operator Hf of second doublet state is the union of three segments, and the discrete spectrum of operator Hf consists of no more than one point. In the two-dimensional case, we have analogous results. In the three-dimensional case, or the essential spectrum of the system in the second doublet state operator Hf is the union of three segments and the discrete spectrum of operator Hf consists of no more than one point, i.e., in the system exists no more than one antibound state, or the essential spectrum of the system in the second doublet state operator Hf is the union of two segments and the discrete spectrum of the operator Hf is empty, or the essential spectrum of the system in the second doublet state operator Hf is consists of a single segment, and discrete spectrum is empty, i.e., in the system the antibound state is absent.

The spectrum of the energy operator of system of four electrons in a crystal described by the Hubbard Hamiltonian in the triplet state were studied in [14]. In the four-electron systems are exists quintet state, and three type triplet states, and two type singlet states. The triplet state corresponds to the basic functions 1 tm n p r = am tap ta+ta+4"o,

tm,n,p,r = am,ta+,tap,lar,t"0, 3tm,n,p,r = am,tar,4-ap,ta+,t

If v =1 and U > 0, then the essential spectrum of the system first triplet state operator 1H1 is exactly the union of two segments and the discrete spectrum of operator 1H1 is empty. In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system first triplet-state operator 1Hi is the union of two segment and the discrete spectrum of operator 1H1 is empty, or the essential spectrum of the system first triplet-state operator 1Hl is single segment and the discrete spectrum of operator 1Hl is empty. If v =1 and U > 0, then the essential spectrum of the system second triplet state operator 2Hl is exactly the union of three segments and the discrete spectrum of operator 2H1 is consists no more than one point.

In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system second triplet-state operator 2H/ is the union of three segments and the discrete spectrum of the operator 2H/ is consists no more than one point, or the essential spectrum of the system second triplet-state operator 2H is the union of two segments and the discrete spectrum of the system second triplet state operator 2H1 is empty, or the essential spectrum of the system second triplet-state operator 2H/ is consists of single segment and the discrete spectrum of the operator 2H/ is empty.

If v = 1 and U > 0, the essential spectrum of the system third triplet-state operator 3H1 is exactly the union of three segments and the discrete spectrum of the operator 3H1 is consists no more than one point. In two-dimensional case, we have analogous results. In the three-dimensional case, the essential spectrum of the system third triplet-state operator 3H1 is the union of three segments, and the discrete spectrum of the operator 3H1 is consists no more than one point or the essential spectrum of the system third triplet-state operator 3H/ is the union of two segments, and the discrete spectrum of the operator 3H/ is empty, or the essential spectrum of the system third triplet-state operator 3H1 is consists of single segment, and the discrete spectrum of the operator 3H1 is empty. We see that there are three triplet states, and they have different origins.

The spectrum of the energy operator of four-electron systems in the Hubbard model in the quintet, and singlet states were studied in [15]. The quintet state corresponds to the free motion of four electrons over the lattice with the basic functions Qmnp-r = am ta+ t< t«vV°. In the work [15] proved, that the spectrum of the system in a quintet state is purely continuous and coincides with the segment [4A — 8Bv, 4A + 8Bv], and the four-electron bound states or the four-electron antibound states is absent. The singlet state corresponds to the basic functions 1s° q r t = a+ta+ta+4a+4-2sp q r t = a+ta+^a+ta+|^°, and these two singlet states have different origins.

If v = 1 and U > 0, then the essential spectrum of the system of first singlet-state operator 1H'l is exactly the union of three segments and the discrete spectrum of the operator is consists only one point. In the two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system first singlet-state operator 1H1 is the union of three segments and the discrete spectrum of the operator 1H'l is consists only one point, or the essential spectrum of the system of first singlet-state operator 1US is the union of two segment and the discrete spectrum of the operator 1H1 is empty, or the essential spectrum of the system of first singlet-state operator 1US is consists of single segment and the discrete spectrum of operator 1US is empty. If v = 1 and U > 0, then the essential spectrum of the system of second singlet-state operator 2H| is exactly the union of three segments and the discrete spectrum of

operator 2H| is consists only one point. In two-dimensional case, we have the analogous results. In the three-dimensional case, the essential spectrum of the system second singlet-state operator 2H| is the union of three segments and the discrete spectrum of the operator 2H| is consists only one point, or the essential spectrum of the system of second singlet-state operator 2H| is the union of two segment and the discrete spectrum of the operator 2H| is empty, or the essential spectrum of the system of second singlet-state operator 2H| is consists of single segment and the discrete spectrum of operator 2H| is empty.

The use of films in various areas of physics and technology arouses great interest in studying a localized impurity state (LIS) of a magnet. Therefore, it is important to study the spectral properties of electron systems in the impurity Hubbard model. The spectrum of the energy operator of three-electron systems in the Impurity Hubbard model in the second doublet state were studied [16]. The structure of essential spectra and discrete spectrum of three-electron systems in the impurity Hubbard model in the Quartet state were studied in [17].

1. Hamiltonian of the system

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 quintet states in the one-dimensional lattice. The Hamiltonian of the chosen model has the form

H = A atn,Yam,Y + B am,Yam+T,y + U ^^ atn^ami,\atn,iami,i + (Ao — A) ^^ a+Ya0,Y +

m, Y m,T, Y m Y

+ (Bo - B) ^(a+YaT,Y + a+Yao,Y) + (Uo - U)a+tao,ta+4ao,|. (2)

T, Y

Here A (Ao) is the electron energy at a regular (impurity) lattice site, B (Bo) is the transfer integral between (between electron and impurities) neighboring sites (we assume that B > 0 (Bo > 0) for convenience), t = ±ej, j = 1, 2,..., v, where ej are unit mutually orthogonal vectors, which means that summation is taken over the nearest neighbors, U (Uo) is the parameter of the on-site Coulomb interaction of two electrons in the regular (impurity) sites, 7 is the spin index, 7 or 7 ^ and ^ denote the spin values 2 and — 1, and amY and am,Y are the respective electron creation and annihilation operators at a site m G .

In the four electron systems has a quintet state, and two type singlet state, and three type triplet states. 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 Smax, Smax 1, ..., Smin, Smax 2 , Smin 0, 1.

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.

2. Four-electron quintet state in the impurity Hubbard model

Let <£>0 be the vacuum vector in the space Has. The quintet state corresponds to the free motion of four electrons over the lattice with the basic functions Vm nkttzv = am ta+ta+ta>0. The subspace H2, corresponding to the quintet state is the set of all vectors of the form = ^m, n, k, lgZ„ f (m,n, k, Z)^ ,n, k, leZ*, f E where is the subspace of antisymmetric functions in the space l 2((Zv)4). We denote by H| the restriction of operator H to the subspace H2. We let = Ao — A, and £2 = Bo — B, and £3 = Uo — U.

Theorem 1. The subspace is invariant under the operator H, and the restriction H| of operator H to the subspace is a bounded self-adjoint operator. It generates a bounded self-adjoint operator H2q acting in the space l^s as

H= 4Af (m, n, k, l) + B ^ [f (m + t, n, k, l) + f (m, n + t, k, l) + f (m, n, k + t, l) +

T

+f (m, n, k, l + t)] + £1 [5™ ,o + 5n ,o + 5k,o + 5Z ,o]f (m, n, k, l) + +£2 [5™ ,of (t, n, k, l)+5n,of (m, t, k, l)+5k, of (m, n, t, l)+5z ,of (m, n, k, t)+5m ,tf (0, n, k, l) +

T

5n,Tf (m, 0, k, l) + 5k,Tf (m,n, 0,l) + 5Z ,tf (m,n,k, 0)]. (3)

The operator H| acts on a vector E as

Hf^ = £ (H 2f )(m,n,k,l)gmin,k,iezv. (4)

m i n i k i ZeZv

Proof. We act with the Hamiltonian H on vectors ^E H2 using the standard anticommutation relations between electron creation and annihilation operators at lattice sites, {amY, a+ p} = 5m , n5Y, {am ,Y, an,p} = {am Y, a+ p} = 0, and also take into account that am, Y<£>o = 0, where 0 is the zero element of H2. This yields the statement of the theorem. □

Lemma 1. The spectra of the operators H| and H2 coincide.

Proof. Because the operators H| and H2 are bounded self-adjoint operators, it follows that if A E ^(H|), then the Weyl criterion ([9], chapter VII, paragraph 3, pp. 262- 263) implies that there is a sequence such that ||^n|| = 1 and

||(Hf — A)^n|| = 0. We set ^ = Y,P,r,t,k /n(P,r,t, k)a+ta+ta+ta+t^o. Then ||(Hf — A)^n||2 = ((Hf — A)^n, (H — A)^n) = E ||(H2 — A)/n(p,r,t, k)||2x

P,r,t,k

x (a+taita+ta+t^o, a+ta+ta+ta+t^o) = Y1 || (H2 — A)Fn(P r,t,k)|Tx

P,r,t,k

x(ak,tat,tar,tap,ta+ta+ta+ta+t^o, <Po) = ^ 11(H^ — A)Fn(p, r, t, k)||2(^o, <Po) =

P,r,t,k

= Ep,r,t,k ||(H 2 — A)Fn(p,r, t, k)||2 ^ 0, as n ^ to, where Fn = ^p,r,t,k /n(p, r, t, k). It follows that A G 0"(H). Consequently, o"(Hf) C o"(H2).

Conversely, let A G o"(H2). Then, by the Weyl criterion, there is a sequence {Fn}^=1 such that ||Fn|| = 1 and lim^^ ||(H — A)^n|| = 0. Setting Fn = ^p,r,t,k /n(p,r,t, k), ||Fn|| = (EP,r,t,k |/n(P,r,t, k)|2)2, we conclude that ||^n|| = ||FT!J| = 1 and ||(H2 — A)Fn|| = ||(H2 — A)^n|| ^ 0 as n ^ to. This means that A G o"(Hf) and hence o"(H2) C 0"(Hf). These two relations imply o"(Hf) = CT(H2). □

We call the operator Hf the four-electron quintet state operator in the impurity Hubbard model.

Let F : l2((ZV)4) ^ L2((TV)4) = h2 be the Fourier transform, where TV is the v— dimensional torus endowed with the normalized Lebesgue measure dA, i.e. A(TV) = 1.

We set Hf = F h2f-1. In the quasimomentum representation, the operator H2 acts in the Hilbert space Las((TV)4), where Las is the subspace of antisymmetric functions in L2((T v )4). _

Theorem 2. The Fourier transform of operator H2 is an operator Hf = FH2F-1 acting in the space Las((TV)4) be the formula

Hf ^f = h(A,^, y, 0)/(A,^, y, 0)+ei[y /(s,^, y, 0)ds+y /(A,t, y, 0)dt+y /(A,^,k, 0)dk+

/„ V „ V

/(A,^, y, C)d£] + 2^[ / ^[cos Ai + cos si]/(s,^, y, 0)ds + / ^[cos ^ + cos ti]x

i=i i=i TV TV TV

V „ V

x/(A,t, y, 0)dt + / ^[cos Yi + cos ki]/(A,^,k, 0)dk + / ^[cos0i + cosCi]/(A,^, y,£)d£,

(5)

where h(A, y, 0) = 4A + 2B ^V=1[c°sAi + cos^i + cosYi + cos0i], and Las is the subspace of antisymmetric functions in L2((TV)4).

Proof. The proof is by direct calculation in which we use the Fourier transformation in formula (3). □

In the impurity Hubbard model, the spectral properties of the considered operator of the energy of four-electron systems are closely related to those of its two-particle subsystems (one-electron systems with impurity). Therefore, we first study the spectrum and localized impurity electron states of the one-electron impurity systems.

3. One-electron systems in the impurity Hubbard model

The Hamiltonian of a one-electron systems in the impurity Hubbard model also has form (2). We let Hi denote the space of one-electron states of the operator H. It is clear that the space H1 is also invariant under operator H. We let H1 denote the restriction of H to the space H1.

Theorem 3. The space H1 is a invariant under the operator H, and the restriction H1 of operator H to the subspace H1 is a bounded self-adjoint operator. It generates a bounded self-adjoint operator H1, acting in the space lfs as

(H1 f )(p) = Af (p) + B ^ f (p + t) + ^p ,o f (p) + ,o f (t) + 5p ,rf (0)], (6)

T T

where 5k, j is the Kronecker symbol. The operator H1 acts on a vector ^ G H1 as

H1 ^ = f )(p)a+t^o. (7)

p , T

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

We let F denote the Fourier transform: F : l2(ZV) ^ L2(TV). We set H1 = FH1F-1. In the quasimomentum representation the operator H1 acts in the Hilbert space L2(TV).

Theorem 4. The Fourier transform of the operator H1 is an bounded self-adjoint operator H1 acting in the space H1 be the formula

V „ „ V

(Hf )(A) = [A + 2B ^ cos Af (A) + eW f (s)ds + 2e2 / ^[cos A* + cos sf (s)ds. (8)

i — 1 i—1

6 — j- T V T V —

Comparing the formulas (5) and (8), and using tensor products of Hilbert spaces and tensor products of operators in Hilbert spaces [17], and taking into account that the function f (A, y, 0) is an antisymmetric function, we can verify that the operator H| can be represented in the form

Hq = H0 i 0 I 0 I+I 0 HH10 I 0 I+I 0 I 0 H10 I+I 0 I 0 I 0 H1,

(9)

where I is the unit operator in space H1.

It is known that the continuous spectrum of operator H fills the entire segment [m Mv] = [A - 2Bv, A + 2Bv].

We set (z) = (b2 - 6s)v-1{ai[b2 + (v - 1)b3] - va2bi}, where

f [ei + cos si]dsi ...dsv _ /cos sfi + 2^ cos Si]dsi...ds„

®1 = 1 + -A—, o n -, ®2 =

A + 2B EV—1 cos Sj — z ' 2 J A + 2B EV—1 cos Sj — z

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TV TV

/" ds1...ds v , , , 0 f cos sjds1...ds v

b1 = 2e2 A I onrv-, b2 = 1 + 2e2 A , o D V-,

J A + 2B j—1 cos Sj — z J A + 2B ,—1 cos Sj — z

TV TV

f cos sjds1...ds V

o3 = 2e2 —--, and v is lattice dimensionality.

J A + 2B ^—1 cos Sj — z

TV

Lemma 3. A number z0 G [mV, MV] is an eigenvalue of operator H1 if and only if it is a zero of the function DV (z).

Proof. Let the number z = z0 G [mV, MV] be an eigenvalue of the operator H^ and <£>(x) be the corresponding eigenfunction, i.e.,

V „ „ V

[A + 2B ^^cosAj]^(A) + eW <£>(S)dS + 2e2 / ^^[cosAj + cosSj]^(S)dS = z0^(A).

Let -0(x) = {A + 2B EV—1 cos Aj — z}<£>(x). Then

u-,, f e1 + 2e2 EV—1[cosAj + cosSj] ,/ w n

W + / -, , ou j—v-^(S)dS = 0,

J A + 2B ^j—1 cos Sj — z

TV

i.e., the number ^ = 1 is a eigenvalue of the operator

+ 2e2 E iV—1[c°SAj + co A + 2B J]V—1 cos Sj — z

\ f ei + 2e2 EV=i[cosAi + coss-] If ^ > K(z) = - ^^^ v-^(s)dsi...dsv.

TV

It then follows that DV(z0) = 0.

Now let z = z0 be a zero of the function DV(z), i.e, DV(z0) = 0. It follows from the Fredholm theorem that the homogeneous equation

If-,, f e1 + 2e2 E V—1[cos Aj + cos Sj] If-,, n

W + -A , nn Uv-^(S)dS = 0

J A + 2B ,—1 cos Sj — z

TV

has a nontrivial solution. This means that the number z = z0 is an eigenvalue of the operator H1. □

Definition 1. The eigenfunction <£> G L2(TV) of the operator H1 corresponding to an eigenvalue z G [mV,MV] is called an local impurity state (LIS) of H^ and z is called the energy of this state.

The following Theorem describe the change of spectrum of the operator H1 in the case v =1.

Theorem 5. A). If = — B and £i < —2B (respectively, = — B and ^ > 2B), then the operator H1 has a unique eigenvalue z = A + lying the below (respectively, above) the continuous spectrum of the operator ii1.

B). If = —2B and e1 < 0 or e2 = 0 and e1 < 0 (respectively, = —2B and e1 > 0 or e2 = 0 and e1 > 0), then the operator HH1 has a unique eigenvalue z = A — a/4B2 + (respectively, z = A + V/4B2 + ^2), lying the below (respectively, above) the continuous spectrum of the operator ii1.

C). If £1 =0 and e2 > 0 (respectively, e1 = 0 and e2 < —2B), then the operator H1 has a unique eigenvalue z = A — 1, (respectively, z = A + JfB—), where E = gf+f^, lying the below (respectively, above) the continuous spectrum of the operator ii1.

D). If e1 = 2(£2+^B£2) (respectively, e1 = — 2(e24f!B£2),) then the operator H1 has a unique eigenvalue z = A + ^f-t^ (respectively, z = A — 2fff-+1)), where E = gf+f^, lying the above (respectively, below) the continuous spectrum of the operator ii1.

E). If e2 > 0 and e1 > 2(e2+f2f£2) (respectively, e2 < —2B and e1 > 2(e2+f2f£2)), then the operator H1 has a unique eigenvalue z1 = A + 2f(a+^2—1~1+a ), where E = gf+f^, and the real number a > 1, lying the above the continuous spectrum of the operator H1.

F). If e2 > 0 and e1 < — 2(e2+f2f£2) (respectively, e2 < —2B and e1 < — 2(e2+f2f£2)), then the operator H1 has a unique eigenvalue z1 = A — 2f(a+^2—1~1+a ), where E = gf+f^,

and the real number a > 1, lying the below the continuous spectrum of the operator ii1.

K). If e2 > 0 and — 2(e2+f2f£2) < e1 < 2(e2+f2f£2) (respectively, e2 < —2B and — 2(£2+2fe2) < < 2(g2+^f£2)), then the operator H1 has a exactly two eigenvalues z1 = A + > M1, and z2 = A + 2f(a-ffV——i+a2) < m1, where E = l+g2,

and real number |a| < 1, lying the above and below the continuous spectrum of the operator

HH1.

M). If —2B < e2 < 0, then the operator H1 has no eigenvalue lying the outside of the continuous spectrum of the operator ii1.

Proof. In the case v =1, the continuous spectrum of the operator H1 coincide with the segment [m1,M1] = [A — 2B, A + 2B]. Expressing all integrals in the equation D1(z) = 0 through the integral J(z) = fT A+2fdc6os s_z, we find that the equation D1(z) = 0 is equivalent to the equation

[^B2 + (e2 + 2B e2)(z — A)] J (z) + (B + ^)2 = 0. (10)

Moreover, the function J(z) is a differentiable function on the set R\ [mx, Mj, in addition, J'(z) = JT [A+2B*C0ss-z]2 > 0, z G [m1,M1]. Thus the function J(z) is an monotone increasing function on (-to, m1) and on (M1, +to). Furthermore, J(z) ^ +0 as z ^ —to, J(z) ^ +to as z ^ m1 — 0, J(z) ^ —to as z ^ M1 + 0, and J(z) ^ —0 as z ^ +to. If e1B2 + (e2 + 2Be2)(z — A) = 0 then from (10) follows that

J (z ) =__<B + £>)2_

E1fl2 + (e| + 2Be2)(z — A) ■ The function ^(z) = — ^B2+(lB_+2Be2)(z-A) has a point of asymptotic discontinuity

z0 = A — If+l^• Since ^'(z) = [£liBB++a+(2l+22)Bz-A)]2 for al1 z = z0 it follows that

the function ^>(z) is an monotone increasing (decreasing) function on (—to,z0) and on (z0, +to) in the case ^ + 2Be2 > 0 (respectively, ^ + 2Be2 < 0), in addition, and if e2 > 0, or e2 < —2B, then ^>(z) ^ +0 as z ^ —to, ^(z) ^ +to as z ^ z0 — 0, ^(z) ^ —to as z ^ z0 + 0, ^(z) ^ —0 as z ^ +to (respectively, if —2B < e2 < 0, then ^(z) ^ —0 as z ^ —to, ^(z) ^ —to as z ^ z0 — 0, ^>(z) ^ +to as z ^ z0 + 0, ^(z) ^ +0 as z ^ +to).

A). If e2 = — B and e1 < —2B (respectively, e2 = — B and e1 > 2B), then the equation for eigenvalues and eigenfunctions (10) has the form

(£1B2 — B2(z — A)} J (z) = 0. (11)

It is clear, that J(z) = 0 for the values z G ^cont(Hi1). Therefore, e1 — z + A = 0, i.e., z = A + e1. If e1 < —2B, then this eigenvalue lying the below of continuous spectrum of the operator ii1, if e1 > 2B, then this eigenvalue lying the above of continuous spectrum of the operator ii1.

B). If e2 = —2B and e1 < 0 (respectively, e2 = —2B and e1 > 0), then the equation for the eigenvalues and eigenfunctions has the form e1B2 J(z) + B2 = 0, that is, J(z) = —. It is clear, what the integral J(z) calculated in a quadrature, of the below (above) of continuous spectrum of the operator ii1, the integral J(z) > 0, (J(z) < 0,) consequently, e1 < 0 (e1 > 0.) The calculated the integral J(z) = fTv A+2Bdcsos s_z, the below of continuous spectrum of the operator H, we have the equation of the form

, 1 == = — —. This equation has a solution z = A — \/e2 + 4B2, lying the below of

yj (A-z)2-4B2 Sl i J o

continuous spectrum of the operator ii1. In the above of continuous spectrum of the operator H1, the equation take the form--, 1 = = — —. This equation has a

^ 1 ^ y/(z-A)2-4B2 £l H

solution of the form z = A + a/e1 + 4B2, lying the above of continuous spectrum of the operator H1.

C). If e1 = 0 and e2 > 0 (respectively, e1 = 0 and e2 < — 2B), then the equation for the eigenvalues and eigenfunctions take in the form

(^ + 2B£2)(z - A) J(z) = — (B + £2)2, or J(z) =

(B + ^

(e2 + 2B^2)(z — A)

Denote E = (f+2'B• Then J(z) = — z—A, or J(z) = a—I • In the below of continuous spectrum of the operator H^, we have the equation of the form = a——z • This

equation has a solution z = A — JEB-1 • It is obviously, that E2 > 1. This eigenvalue lying the below of continuous spectrum of the operator H^. In the above of continuous spectrum of the operator H/i, the equation for the eigenvalues and eigenfunctions has the form--, 1 = = —^r. From here, we find z = A +—,2B2E . This eigenvalue lying

(z—A)2—4B2 z—A ' VE2—1 6 J 6

the above of continuous spectrum of the operator H/1.

D). If e1 = 2(£2+^B£2), then the equation for eigenvalues and eigenfunctions has the form (e2 + 2Be2)(z — A + 2B) J(z) = —(B + e2)2, from this

J (z ) = — (5+2bB)+7—ÂT2B). (12)

We denote E = 'fj+fl^. In the first we consider the equation (12) in the below of continuous spectrum of the operator H/1. In equation (12) we find the equation of the form , 1 == = —. From this, we find z1 = A + 2B(f +1), and z2 = A — 2B. Now

^/(A—z)2 — 4B2 A—z—2B ' 1 1 £2 —1 ' 2

we verify the conditions z^ < A — 2B, i = 1, 2. The inequality z1 < A — 2B, is incorrectly, and inequality z2 < A — 2B, is incorrectly. We now consider the equation (12) in the

1E

above of continuous spectrum of the operator Hi. We have--, 1 = = - . ,„„.

1 1 1 yj (z—A)2—4B2 z-A+2f

From this equation we find the same solutions z1 and z2, the outside of the domain of continuous spectrum of the operator ii1. Now we verify the conditions z^ > A+2B, i = 1, 2. The inequality z1 > A + 2B, is correctly, and inequality z2 > A + 2B, is incorrectly. Consequently, in this case the operator H has a unique eigenvalue z1 = A + 2f^f_++1), lying the above of continuous spectrum of the operator ii1.

Let e1 = — 2(e2+^f£2), then the equation of eigenvalues and eigenfunctions take in the form J(z) = — z—A—2b , where E = f+g. _

In the below of continuous spectrum of the operator ii1, we have equation of the form , 1 == = —^tb . From here we find zi = A — ^ff +1), and z2 = A + 2B.

yj (A—z)2—4B2 A-z+2B 1 f2 — 1 ' 2 1

The appear inequalities z1 < A — 2B, is correct, and z2 < A — 2B, is incorrect. In the above of continuous spectrum of the operator ii1, we have equation of the form

1 = —2B. It follows that, what z1 = A — , and z2 = A + 2B.

^/(z—A)2— 4B2 z—A—2B' ^ ^ ^ s2 —1

The inequality > A + 2B, and z2 > A + 2B, are incorrectly. Therefore, in this case the operator H1 has a unique eigenvalue = A — 2BEf_ +1), lying the below of continuous spectrum of the operator i^.

E). If 52 > 0 and 51 > 2(£_4B2B£_), (respectively, 52 < —2B and 51 > 2(£_4B2B£_)), then consider necessary, that = a x 2(£2+|B£2), where a > 1— real number. Then the equation for eigenvalues and eigenfunctions has the form

{a x 2(52 + 2B£2) x B2 + (e2 + 2B52)(z — A)}J(z) + (B + 52)2 = 0, B

or (e2 + 2B52)(z — A + 2aB) J(z) + (B + 52)2 = 0. From this

(B + 52)2

J (z) =

(52 + 2B 52)(z — A + 2aB)' We denote E = £B+£_£_, then J(z) = — z-A+2aB. In the first we consider this equation in the below of continuous spectrum of the operator H1. Then , 1 == = —D. This equation has the solutions

„ 2B (a + E VE2 — 1 + a2) , „ 2B (a — E VE2 — 1 + a2)

z1 = A +-------, and z2 = A +-------.

1 E2 — 1 ' 2 E2 — 1

Now, we verify the condition z^ < A — 2B,i = 1,2. The solution z1 no satisfy the condition z1 < A — 2B, but z2 satisfy the condition z2 < A — 2B. We now verify the conditions z2 < A — 2aB. The appear, this inequality is incorrectly. The appear inequalities z1 > A + 2B is correct, and z2 > A + 2B, is incorrect. We now verify the conditions z1 > A — 2aB. So far as, A — 2aB < A + 2B, the appear, this inequality is correctly. Consequently, in this case, the operator H has a unique eigenvalue z1 = A + 2B(a+EE2E1-1+a ), above of continuous spectrum of the operator ii1.

F). If 52 > 0, and 51 < — 2(£_ +B2B£_), (respectively, £2 < — 2B, and £1 < — 2(£_+B2B£_)), then we assume that 51 = —a x 2(£_ +|B£2), where a > 1— real number. The equation for eigenvalues and eigenfunctions take in the form

(52 + 2B 52)(z — A — 2aB) J (z) = —(B + 52)2.

From here J(z) = — (£2+2B(£B2)"+££_)A_-2aB). The introduce notation E = . Then

E

J<z> = — z — A — 2aB • <13»

In the below of continuous spectrum of the operator ii1, we have the equation

J(z) = —-, from here

A - z + 2aB ^(A - z)2 - 4B2 A - z + 2aB

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This equation take the form

(E2 - 1)(A - z)2 - 4aB(A - z) - 4B2(E2 + a2) = 0.

We find

„ 2B (a + EVE2 - 1 + a2) , , 2B (a - EVE2 - 1 + a2)

zi = A--1-and z2 = A--E231-.

We now verify the conditions z^ < m1 = A - 2B,i = 1, 2. The appear, that z1 < A - 2B, is correctly and z2 < A - 2B, is incorrectly. Now we consider the equation (13) in the above of continuous spectrum of the operator ii1. Then J(z) = - 2aB. From this --/ \ =--jVr. We find

, 2B(a + EVE2 - 1 + a2) , , 2B(-a + EVE2 - 1 + a2)

z1 = A--—-, and z2 = A +--—-.

1 E2 - 1 ' 2 E2 - 1

We verify the conditions z^ > A + 2B, i = 1, 2. The appear z1 > A + 2B, it is not true,

and the z2 > A + 2B, is true. We now verify the conditions z2 > A + 2aB. The appear,

this inequality is incorrectly. Consequently, in this case, the operator H have unique

eigenvalue z1 = A - 2B(a+TT2T1-1+a ) < m1, i.e., lying the below of continuous spectrum

of the operator H1.

0 -^--- C1 < -w

K). If £2 > 0 and -2(£2+2B£2) < £1 < 2(£2+2B£2) (respectively, £2 < -2B and

b ^ ^ £ 2+2B£ 2)), the we take 51 = a x ^^ B real number. Then the equation for eigenvalues and eigenfunctions has the form

2(£2+2B£2) < £1 < 2(£2+b2B£2)), the we take £1 = a x 2(£2+R2B£2), where -1 < a < 1-

(52 + 2B52)(z — A + 2aB) J(z) = —(B + 52)2, |a| < 1. (14)

We denote E = £B+2b£2 . Then the equation (14) receive the form

J(z) = E

z — A + 2aB'

In the below of continuous spectrum of the operator ii1 we have equation of the form

1E

, |a| < 1.

a/(A - z)2 - 4B2 A - z - 2aB

This equation has a solutions

„ 2B (a + EVE2 - 1 + a2) , „ 2B (a - EVE2 - 1 + a2)

z1 = A +--^----, and z2 = A +--^----.

^ E2 - 1 ' 2 E2 - 1

The inequalities z1 < A - 2B, and z1 < A - 2aB, is incorrectly. The inequalities

z2 < A - 2B, is correctly. We now verify the conditions z2 < A - 2aB, since

A — 2B < A — 2aB, this inequality is true. We now consider the equation (14) in the above of continuous spectrum of the operator ii1. We have the equation of the form

1E

sj (z — A)2 — 4B2 z — A + 2aB' This equation has a solutions

„ 2B (a + E VE2 — 1 + a2 ) , „ 2B (a — E VE2 — 1 + a2 )

z1 = A +-------, and z2 = A +-------.

1 E2 — 1 ' 2 E2 — 1

The inequalities z1 > A + 2B, and z1 > A — 2aB is true, as A + 2B > A — 2aB, that the inequality z1 > A — 2aB is correctly. The inequalities z2 > A + 2B, and z2 > A + 2aB is incorrectly. Consequently, in this case the operator H/1 has a exactly two eigenvalues

, 2B (a + E VE2 — 1 + a2 ) , , 2B (a — E VE2 — 1 + a2 )

z1 = A +----\--, and z2 = A +-------,

1 E2 — 1 ' 2 E2 — 1 '

lying the above and below of continuous spectrum of the operator H/1.

M). If —2B < e2 < 0, then e2+2Be2 < 0, and the function ^(z) = — £ib+JB+B^X^A) is a decreasing function in the intervals (—to, z0) and (z0, +to); By, z ^ —to the function ■0(z) ^ —0, and by z ^ z0 — 0, the function ^(z) ^ —to, and by z ^ +to, ^(z) ^ +0, and by z ^ z0 + 0, ^(z) ^ +to. The function J (z) ^ 0, by z ^ —to, and by z ^ m1 — 0, the function J (z) ^ +to, and by z ^ M1 + 0, the function J (z) ^ —to, by z ^ +to, the function J (z) ^ —0. Therefore, the equation ^>(z ) = J (z ), that's impossible the solutions in the outside the continuous spectrum of operator H/1. Therefore, in this case, the operator H1 has no eigenvalues lying the outside of continuous spectrum of the operator H1. □

Consequently, the spectrum of operator H/1 is consists from continuous spectrum and at most two eigenvalues.

4. Main results of the work

Now, using the obtained results and representation (9), we describe the structure of essential spectrum and discrete spectrum of the energy operator of four electron systems in the impurity Hubbard model in the quintet state.

Theorem 6. Let v =1. Then

A). If £2 = — B and < —2B (respectively, £2 = — B and £x > 2B), then the essential spectrum of the operator Hf is consists of the union of four segments:

o-eee(H|) = [4A — 8B, 4A + 8B] U [3A — 6B + z, 3A + 6B + z]U