# Correct Singular Perturbations of the Laplace Operator with the Spectrum of the Unperturbed Operator

## TL;DR

This paper investigates the spectral properties of the Laplace operator with a singular potential perturbation, extending previous self-adjoint studies to non-self-adjoint cases using a new analytical method.

## Contribution

It introduces a novel method for analyzing non-self-adjoint singular perturbations of the Laplace operator, expanding the understanding of spectral issues in such contexts.

## Key findings

- Spectral properties are characterized for non-self-adjoint singular perturbations.
- A new analytical approach is developed for these perturbations.
- Results extend previous self-adjoint spectral analysis to more general cases.

## Abstract

The work is devoted to the study of Laplace operator when the potential is a singular generalized function and plays the role of a singular perturbation of a Laplace operator. Abstract theorem obtained earlier by the authors B.N. Biyarov and G.K. Abdrasheva applies to this. The main purpose of the study is the spectral issue. Singular perturbations for differential operators have been studied by many authors for the mathematical substantiation of solvable models of quantum mechanics, atomic physics, and solid state physics. In all these cases, the problems were self-adjoint. In this paper, we consider non-self-adjoint singular perturbation problems. A new method has been developed that allows investigating the considered problems.