Asymptotic behaviour of the spectrum of a waveguide with distant perturbations

TL;DR

This paper analyzes how the eigenvalues and eigenfunctions of a waveguide's Dirichlet Laplacian change asymptotically when two distant perturbations are introduced, providing convergence results and explicit asymptotic expansions.

Contribution

It presents new asymptotic formulas for the spectrum of a waveguide with distant perturbations, including convergence theorems and eigenvalue expansions.

Findings

Eigenvalues converge as perturbations become distant

Explicit asymptotic expansions for eigenvalues are derived

Eigenfunctions' asymptotic behavior is characterized

Abstract

We consider the waveguide modelled by a $n$-dimensional infinite tube. The operator we study is the Dirichlet Laplacian perturbed by two distant perturbations. The perturbations are described by arbitrary abstract operators ''localized'' in a certain sense, and the distance between their ''supports'' tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such system. The main results are a convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some particular examples of the distant perturbations. The examples are the potential, second order differential operator, magnetic Schroedinger operator, curved and deformed waveguide, delta interaction, and integral operator.