A spectral shift function for Schr\"{o}dinger operators with singular interactions

TL;DR

This paper derives an explicit spectral shift function for Schrödinger operators with singular delta interactions supported on smooth hypersurfaces, using Dirichlet-to-Neumann maps, advancing understanding of spectral changes due to such singular potentials.

Contribution

It provides an explicit formula for the spectral shift function for Schrödinger operators with delta interactions on hypersurfaces, utilizing Dirichlet-to-Neumann maps.

Findings

Explicit spectral shift function derived

Applicable to operators with hypersurface-supported delta potentials

Enhances spectral analysis of singular interactions

Abstract

For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is determined in an explicit form with the help of (energy parameter dependent) Dirichlet-to-Neumann maps. Here $\delta_{\cal{C}}$ denotes a singular $\delta$-potential which is supported on a smooth compact hypersurface $\mathcal{C}\subset\mathbb{R}^n$ and $\alpha$ is a real-valued function on $\mathcal{C}$.