Distant perturbations of the Laplacian in a multi-dimensional space

TL;DR

This paper investigates how the discrete spectrum of the Laplacian in multi-dimensional space is affected by distant localized perturbations, providing asymptotic expansions and convergence results as the perturbations become infinitely separated.

Contribution

It introduces a general framework for analyzing the asymptotic behavior of the Laplacian's spectrum under various distant perturbations, including potential, differential, magnetic, and integral operators.

Findings

Convergence theorem for the spectrum as perturbations tend to infinity

Asymptotic expansions for eigenelements in the presence of distant perturbations

Examples demonstrating different types of perturbations and their spectral effects

Abstract

We consider the Laplacian in $\mathbb{R}^n$ perturbed by a finite number of distant perturbations those are abstract localized operators. We study the asymptotic behaviour of the discrete spectrum as the distances between perturbations tend to infinity. The main results are the convergence theorem and the asymptotics expansions for the eigenelements. Some examples of the possible distant perturbations are given; they are potential, second order differential operator, magnetic Schrodinger operator, integral operator, and $\d$-potential.