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

Abstract

We study the Laplacian in $L^2(\mathbb{R}^3)$ perturbed on an infinite curve $Γ$ by a $δ$ interaction defined through boundary conditions which relate the corresponding generalized boundary values. We show that if $Γ$ 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.