Eigenvalues for a Schrodinger operator on a closed Riemannian manifold with holes

TL;DR

This paper investigates how the eigenvalues of a Schrödinger operator on a closed Riemannian manifold are affected when holes are introduced, comparing the original and perforated manifolds with Dirichlet boundary conditions.

Contribution

It provides a comparison framework for eigenvalues of Schrödinger operators on manifolds before and after introducing holes, extending spectral analysis to perforated geometries.

Findings

Eigenvalues decrease when holes are introduced

Comparison formulas relate eigenvalues before and after perforation

Results applicable to spectral geometry and quantum mechanics

Abstract

In this article we consider a closed Riemannian manifold (M,g) and A a subset of M. The purpose of this article is the comparison between the eigenvalues of a Schrodinger operator on the manifold (M,g) and the eigenvalues on the manifold (M-A,g) with Dirichlet boundary conditions.