On the localization of the spectrum of some perturbations of a two-dimensional harmonic oscillator

Abstract

In this paper, we study the localization of the discrete spectrum of certain perturbations of a two-dimensional harmonic oscillator. The convergence of the expansion of the source function in terms of the eigenfunctions of a two-dimensional harmonic oscillator is investigated. A representation of Green's function of a two-dimensional harmonic oscillator is obtained. The singularities of Green's function are highlighted. The well-posed definition of the maximal operator generated by a two-dimensional harmonic oscillator on a specially extended domain of definition is given. Then, we describe everywhere solvable invertible restrictions of the maximal operator. We establish that the eigenvalues of a harmonic oscillator will also be the eigenvalues of well-posed restrictions. The results are supported by illustrative examples.

Keywords:

• Green's functions for elliptic equations
• boundary value problems for second-order elliptic equations
• harmonic analysis of several complex variables
• meromorphic functions of several complex variables
• estimates of eigenvalues in the context of PDEs
• asymptotic distributions of eigenvalues in context of PDEs

AMS SUBJECT CLASSIFICATIONS:

• 31A25
• 35J08
• 35J25
• 32A50
• 32A20
• 35P15
• 35P20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the Ministry of Science and Education of the Republic of Kazakhstan [grant numbers AP08855402].The second author is supported by the Development Program of the Scientific and Educational Mathematical Center of the Volga Federal District, additional agreement no. 075-02-2020-1421/1 to agreement no. 075-02-2020-1421.