# Boundary integral formulations of eigenvalue problems for elliptic differential operators with singular interactions and their numerical approximation by boundary element methods

## TL;DR

This paper develops boundary integral formulations for eigenvalue problems involving elliptic differential operators with singular interactions, enabling numerical approximation via boundary element methods with proven convergence.

## Contribution

It introduces new boundary integral formulations for elliptic operators with singular interactions and demonstrates their suitability for numerical eigenvalue computation.

## Key findings

- Self-adjointness of operators with singular interactions established
- Boundary element methods effectively approximate eigenvalues and eigenfunctions
- Convergence of numerical methods validated with numerical examples

## Abstract

In this paper the discrete eigenvalues of elliptic second order differential operators in $L^2(\mathbb{R}^n)$, $n \in \mathbb{N}$, with singular $\delta$- and $\delta'$-interactions are studied. We show the self-adjointness of these operators and derive equivalent formulations for the eigenvalue problems involving boundary integral operators. These formulations are suitable for the numerical computations of the discrete eigenvalues and the corresponding eigenfunctions by boundary element methods. We provide convergence results and show numerical examples.