AIP conference proceedings

Spectrum and resolvent of the operator matrix associated to a system describing at most four quantum particles on a lattice

Tulkin Husenovich Rasulov, Elyor B. Dilmurodov, Hakimboy M. Latipov

Published Jan 1, 2025• 0 citations

Abstract

This study investigates the tridiagonal operator matrix 𝒜 of order 4, which is defined within the truncated four-particle subspace of the Fock space. The essential spectrum of the operator 𝒜 is derived. The main block components are constructed, and the relationships between their spectra are analyzed. To determine the eigenvalues of the operator matrix 𝒜, the corresponding Fredholm determinant is computed. Additionally, it is demonstrated that the resolvent operator associated with the operator matrix 𝒜 is also represented in matrix form.

DOI: 10.1063/5.0299534

Related papers

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

Sadulla Mamarazhabovich Tashpulatov, R. T. Parmanova

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.

Journal of Applied Mathematics and PhysicsJan 1, 20214 citations

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

Sadulla Mamarazhabovich Tashpulatov, R. T. Parmanova

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.

Journal of Applied Mathematics and PhysicsJan 1, 20230 citations

The first and second Schur complements corresponding to 3 × 3 operator matrix in fermionic fock space

D. E. Ismoilova

In this paper, we examine a 3 × 3 operator matrix 𝒜 within the context of fermionic Fock space. This operator is associated with the energy operator (Hamiltonian). This energy operator is derived from a non-preserved system containing three interacting quantum particles. We portray the location of the σess(𝒜) and construct the first and second Schur complements associated with the investigated operator matrix. We analyze the primary spectral characteristics of these two Schur complements and provide a realization of the well-known Birman-Schwinger principle for the first Schur complement.

AIP conference proceedingsJan 1, 20250 citations

Spectra of the Energy Operator of Two-Electron System in the Impurity Hubbard Model

Sadulla Mamarazhabovich Tashpulatov

We consider two-electron systems for the impurity Hubbard Model and investigate the spectrum of the system in a singlet state for the v-dimensional integer valued lattice Zv. We proved the essential spectrum of the system in the singlet state is consists of union of no more then three intervals, and the discrete spectrum of the system in the singlet state is consists of no more then five eigenvalues. We show that the discrete spectrum of the system in the triplet and singlet states differ from each other. In the singlet state the appear additional two eigenvalues. In the triplet state the discrete spectrum of the system can be empty set, or is consists of one-eigenvalue, or is consists of two eigenvalues, or is consists of three eigenvalues. For investigation the structure of essential spectra and discrete spectrum of the energy operator of two-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 two-electron systems in an impurity Hubbard model.

Journal of Applied Mathematics and PhysicsJan 1, 20228 citations

Coupled-cluster method in Fock space. I. General formalism

Leszek Z. Stolarczyk, Hendrik J. Monkhorst

The problem of finding the spectrum of the Fock-space Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{} for a system of many fermions is analyzed. The quasiparticle formalism is employed, with ``holes'' and ``particles'' defined with respect to some N-particle determinantal wave function (the model vacuum). The basic idea behind the proposed approach is to perform a similarity transformation of Hamiltonian \ifmmode \hat{H}\else \^{H}\fi{}, such that the resulting effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{} is, unlike \ifmmode \hat{H}\else \^{H}\fi{}, a quasiparticle-number--conserving operator. It is shown that eigenvalues of \ifmmode \hat{G}\else \^{G}\fi{}, corresponding to small numbers of quasiparticles (0,1,2) can be easily calculated. This is equivalent to finding eigenvalues of \ifmmode \hat{H}\else \^{H}\fi{} for certain states of N, N\ifmmode\pm\else\textpm\fi{}1, and N\ifmmode\pm\else\textpm\fi{}2 particles. The construction of the operator transforming \ifmmode \hat{H}\else \^{H}\fi{} into \ifmmode \hat{G}\else \^{G}\fi{} (the wave operator) stems from an analysis of the structure of the algebra of linear operators acting in a (finite-dimensional) Fock space. The exponential Ansatz for the wave operator is used, resulting in a generalization of the coupled-cluster (CC) method of Coester [Nucl. Phys. 7, 421 (1958)]. The generalized CC equations determining the wave operator, and equations determining the effective Hamiltonian \ifmmode \hat{G}\else \^{G}\fi{}, are presented in a diagrammatic form. An effort has been made to obtain a concise notation for expressing these equations in an algebraic form. Approximation schemes, necessary for practical applications of the proposed method, are also studied.

Physical Review AAug 1, 1985171 citations

Zernike polynomials from the tridiagonalization of the radial harmonic oscillator in displaced Fock states

Hashim A. Yamani, Zouhaı̈r Mouayn

We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D and n = 0,1,2,.... Remarkably, for fixed n,and varying z in D, the system phi(z)n forms a family of Perelomov-type coherent states associated with the RHO. For each fixed n, the expansion of phi(z)n over the basis (fs) of eigenfunctions of the RHO yields coefficients cn,s(z; z) precisely given by two-dimensional complex Zernike polynomials. The key insight is that the algebraic tridiagonal structure of RHO contains the complete information about the bound state solutions of the two-dimensional Schrödinger operator describing a charged particle in a magnetic field (of strength proportional to B > 1/2) on the Poincaré disc D.

arXiv:2506.22933Jun 28, 20250 citations