Spectra of the Energy Operator of Four-Electron Systems in the Impurity Hubbard Model. Triplet State

We consider the energy operator of four-electron systems in an impurity Hubbard model and investigated the structure of essential spectra and discrete spectrum of the system in the first triplet state in a one-dimensional lattice. 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. The investigations show that there are such cases: 1) the essential spectrum of the system consists of the union of no more than eight segments, and the discrete spectrum of the system consists of no more than three eigenvalues; 2) 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; 3) the essential spectrum of the system consists of the union of no more than three segments, and the discrete spectrum of the system is the empty set. Consequently, 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.


Introduction
In the early 1970s, three papers [1] [2] [3], where a simple model of metal was 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 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 is currently one of the most extensively studied multielectron models of metals [6] [7] [8] [9] [10]. 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 [6] [7] [8] [9] [10]. It was proved in [6] that the spectrum of the system Hamiltonian The spectrum and wave functions of the system of three electrons in a crystal described by the Hubbard Hamiltonian were studied in [11]. The three-electron systems exist quartet state, and two type doublet states.
The spectrum of the energy operator of a system of four electrons in a crystal described by the Hubbard Hamiltonian in the triplet state was studied in [12].
The four-electron systems exist quintet states, and three type triplet states, and two type singlet states. The triplet state corresponds to the basic functions The spectrum of the energy operator of four-electron systems in the Hubbard model in the quintet and singlet states was studied in [13]. The quintet state corresponds to the free motion of four electrons over the lattice with the basic 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 was studied [14].
The structure of essential spectra and discrete spectrum of three-electron systems in the impurity Hubbard model in the Quartet state were studied in [15].

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

Four-Electron First Triplet State in the Impurity Hubbard Model
,0 ,0 ,0 ,   p q r k f p q r k a a a a ψ ϕ  , ,   2  1  1  , , ,   , , ,  , , , , 0, as Conversely, let ( ) We set 1 In the quasimomentum representation, the operator    Proof. The proof is by direct calculation in which we use the Fourier transformation in formula (3).

 Journal of Applied Mathematics and Physics
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). We first study the spectrum and localized impurity electron states of the one-electron impurity systems.

One-Electron Systems in the Impurity Hubbard Model
The Hamiltonian of one-electron systems in the impurity Hubbard model also has form (2). We let 1  denote the space of one-electron states of the operator H. It is clear that the space 1  is also invariant under operator H. We let H 1 denote the restriction of H to the space 1  .
Theorem 3. (coordination representation of the action of operator H 1 ) The space 1  is an invariant under operator H, and restriction H 1 of operator H to the subspace 1  is a bounded self-adjoint operator. It generates a bounded self-adjoint operator 1 H , acting in the space 2 as l as is the Kronecker symbol. The operator H 1 acts on a vector Lemma 2. The spectra of the operators 1 H and 1 H coincide.
We let  denote the Fourier transform: 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 ( ) , , ,   1

B E z A E
1

B E z A E
We denote ( ) In the first, we consider Equation (12)   1 In the below continuous spectrum of the operator 1 H , we have the equation