Spectral asymptotics induced by approaching and diverging planar circles

TL;DR

This paper investigates how the eigenvalues of a 2D quantum system with delta interactions on two concentric circles change as the circles approach each other or move infinitely apart, revealing spectral asymptotic behaviors.

Contribution

It provides new asymptotic analysis of eigenvalues for a 2D Hamiltonian with delta interactions supported on concentric circles as their separation varies.

Findings

Eigenvalues exhibit specific asymptotic behaviors as the circles approach.

Eigenvalues tend to different limits as the circles diverge.

The analysis characterizes spectral shifts due to geometric changes.

Abstract

We consider two dimensional system governed by the Hamiltonian with delta interaction supported by two concentric circles separated by distance $d$. We analyze the asymptotics of the discrete eigenvalues for $d \to 0$ as well as for $d\to \infty$.