The Structure of Essential Spectra and Discrete Spectrum of Three-Electron Systems in the Impurity Hubbard Model—Quartet State

We consider a three-electron system in the Impurity Hubbard model with a coupling between nearest-neighbors. Our research aim consists of studying the structure of essential spectrum and discrete spectra of the energy operator of three-electron systems in the impurity Hubbard model in the quartet state of the system in a v-dimensional lattice. We have reduced the study of the spectrum of the three-electron quartet state operator in the impurity Hubbard model to the study of the spectrum of a simpler operator. We proved the essential spectra of the three-electron systems in the Impurity Hubbard model in the quartet state is the union of no more than six segments, and the discrete spectrum of the system is consists of no more than four eigenvalues.


Introduction
In the early 1970s, three papers [1] [2] [3], where a simple model of 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 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 How to cite this paper: Tashpulatov 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 an approximation used in solid-state physics to describe the transition between conducting and insulating states. It is the simplest model describing particle interaction on a lattice.
The Hubbard model and impurity Hubbard model is currently one of the most extensively studied multielectron models of metals [6] [7] [8] [9] [10]. But little is known about exact results for the spectrum and wave functions of the crystal described by the Hubbard model, and obtaining the corresponding statements is therefore 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 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 0 U < : 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 0 U → 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 [11].
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 [12]. The four-electron system 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 was studied in [13].
Here, we consider the energy operator of three-electron systems in the Impurity Hubbard model and describe the structure of the essential spectra and dis- Here A (A 0 ) is the electron energy at a regular (impurity) lattice site, B (B 0 ) is the transfer integral between (between electron and impurities) neighboring sites (we assume that The three electron systems have a quartet state and two type doublet states.
The energy of the system depends on its total spin S. Along with the Hamiltonian, the e N electron system is characterized by the total spin S, Hamiltonian (1) The proof of Theorem 2, is straightforward of (2) using the Fourier transformation.
It is clear that spectral properties of energy operator of three-electron systems in the impurity Hubbard model in the quartet state are closely related to the spectral properties of its one-electron subsystems in the impurity Hubbard model. First we investigate the spectrum of one-electron subsystems.

Spectra of the Energy Operator of One-Electron System in the Impurity Hubbard Model
The Hamiltonian H of one-electron systems in the impurity Hubbard model also has the form (1). We let As in the proof of Theorem 3, using the standard anticommutation relations between electron creation and annihilation operators at lattice sites, we get the following.
Suppose first that 1 ν = and denote Now substitute (9) in expressing of a and b we get the following system of two Therefore, it is true the following.
The following Theorem describe of the exchange of the spectrum of operator 1 H in the case 1 ν = .
The function calculated in a quadrature, of the below (above) of continuous spectrum of op- The calculated the integral ( ) This equation has a solution of the form The calculated the integral ( ) This equation has a solution This equation has a solution This eigenvalue lying the below of the continuous spectrum of operator From here, we find In the first we consider the Equation (25) , also is incorrectly. We now consider the Equation (25) in the above of continuous spectrum of operator In this equation we find the solutions above of continuous spectrum of operator 1 H . Now we verify the conditions 2 , 1, 2 In the below of continuous spectrum of operator 1 H , we have equation of the form ( ) From here we find ( ) The appear in- In the first we consider this equation in the below of the continuous spectrum This equation has the solutions ( ) . We now verify the conditions The introduce notation ( ) In the below of the continuous spectrum of operator 1 H , we have the equa- This equation take the form We find ( ) This equation has a solutions ( ) , is correctly, and the      (12)   i.e., 1 from this we have the equation in the form (14): In the first we consider the Equation (14) in the below of continuous spectrum of operator 1 H .
In the below of continuous spectrum of operator In the first we consider this equation in the below of the continuous spectrum of operator 1 has a unique solution, if ( ) . This inequality is incorrect. Therefore, the below of continuous spectrum of operator 1 H , the operator 1 H has no eigenvalues.
The above of continuous spectrum of operator 1 H , we have the ( ) The equation . This inequality is correctly.
Therefore, the below of continuous spectrum of operator 1 H , the operator 1 H has a unique eigenvalues.