Hiatus perturbation for a singular Schr\"odinger operator with an interaction supported by a curve in \mathbb{R}^3

TL;DR

This paper studies the spectral properties of Schrödinger operators with singular interactions supported by curves in three-dimensional space, focusing on how small gaps in the curve affect eigenvalues and deriving asymptotic expansions.

Contribution

It provides a rigorous definition of such operators, analyzes conditions for the existence of discrete spectrum, and derives asymptotic behavior of eigenvalues with respect to curve perturbations.

Findings

Discrete spectrum can be empty for short curves.

Eigenvalues exhibit asymptotic behavior involving epsilon log epsilon.

Asymptotic expansion describes eigenvalue shifts due to curve hiatus.

Abstract

We consider Schr\"odinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a finite curve $\Gamma$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $\Gamma$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2\epsilon$. We derive an asymptotic expansion with the leading term which a multiple of $\epsilon \ln\epsilon$.