Strong-coupling asymptotic expansion for Schr\"odinger operators with a singular interaction supported by a curve in $\mathbb{R}^3$

TL;DR

This paper derives strong-coupling asymptotic expansions for the discrete spectrum of Schrödinger operators with singular interactions supported by curves in three-dimensional space, revealing spectral gap formation for non-straight periodic curves.

Contribution

It provides new asymptotic formulas for the spectrum of Schrödinger operators with curve-supported singular interactions, including cases of loops and periodic curves, linking geometry to spectral properties.

Findings

Asymptotic expansion of discrete spectrum for strong coupling

Spectral gaps appear for non-straight periodic curves

Spectrum behavior depends on curvature of the supporting curve

Abstract

We investigate a class of generalized Schr\"{o}dinger operators in $L^2(\mathbb{R}^3)$ with a singular interaction supported by a smooth curve $\Gamma$. We find a strong-coupling asymptotic expansion of the discrete spectrum in case when $\Gamma$ is a loop or an infinite bent curve which is asymptotically straight. It is given in terms of an auxiliary one-dimensional Schr\"{o}dinger operator with a potential determined by the curvature of $\Gamma$. In the same way we obtain an asymptotics of spectral bands for a periodic curve. In particular, the spectrum is shown to have open gaps in this case if $\Gamma$ is not a straight line and the singular interaction is strong enough.