Spectral theory for Schr\"odinger operators with $\delta$-interactions supported on curves in $\mathbb R^3$

TL;DR

This paper develops a comprehensive spectral and scattering theory for Schr"odinger operators with delta interactions supported on curves in three-dimensional space, including eigenvalue bounds, inequalities, and scattering matrix representations.

Contribution

It introduces a systematic framework for analyzing Schr"odinger operators with delta interactions on curves, including new bounds, inequalities, and explicit scattering matrix formulas.

Findings

Bounds for the number of negative eigenvalues based on curve geometry

An isoperimetric inequality for the principal eigenvalue

Explicit representation of the scattering matrix

Abstract

The main objective of this paper is to systematically develop a spectral and scattering theory for selfadjoint Schr\"odinger operators with $\delta$-interactions supported on closed curves in $\mathbb R^3$. We provide bounds for the number of negative eigenvalues depending on the geometry of the curve, prove an isoperimetric inequality for the principal eigenvalue, derive Schatten--von Neumann properties for the resolvent difference with the free Laplacian, and establish an explicit representation for the scattering matrix.