Curvature-induced bound states for a $\delta$ interaction supported by a curve in $\mathbb{R}^3$

TL;DR

This paper investigates how the curvature of an infinite smooth curve in three-dimensional space induces bound states in the Laplacian operator with a delta interaction supported on the curve, revealing conditions for the existence of eigenvalues below the essential spectrum.

Contribution

It demonstrates that curvature in a non-straight, asymptotically straight curve supports bound states in the Laplacian with delta interactions, extending understanding of geometric effects on quantum operators.

Findings

Curvature induces at least one bound state below the essential spectrum.

Asymptotic straightness of the curve is sufficient for bound state existence.

The delta interaction's invariance along the curve is a key condition.

Abstract

We study the Laplacian in $L^2(\mathbb{R}^3)$ perturbed on an infinite curve $\Gamma$ by a $\delta$ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if $\Gamma$ is smooth and not a straight line but it is asymptotically straight in a suitable sense, and if the interaction does not vary along the curve, the perturbed operator has at least one isolated eigenvalue below the threshold of the essential spectrum.