Asymptotics of eigenvalues for an energy operator of the one model of quantum physics

TL;DR

This paper analyzes the asymptotic behavior of eigenvalues for an energy operator in a quantum physics model involving a two-level system and harmonic oscillator, using advanced matrix techniques.

Contribution

It introduces a modified diagonalization method and new compactness criteria to derive eigenvalue asymptotics for the model's energy operator.

Findings

First two terms of eigenvalue asymptotics obtained

Remainder estimates provided

Eigenvalues relate to perturbed harmonic oscillator spectrum

Abstract

In this paper we consider eigenvalues asymptotics of the energy operator in the one of the most interesting models of quantum physics, describing an interaction between two-level system and harmonic oscillator. The energy operator of this model can be reduced to some class of infinite Jacobi matrices. Discrete spectrum of this class of operators represents the perturbed spectrum of harmonic oscillator. The perturbation is an unbounded operator compact with respect to unperturbed one. We use slightly modified Janas-Naboko successive diagonalization approach and some new compactness criteria for infinite matrices. Two first terms of eigenvalues asymptotics and the estimation of remainder are found.