Magnetic field influence on the discrete spectrum of locally deformed leaky wires

TL;DR

This paper investigates how a magnetic field affects the discrete spectrum of a Schrödinger operator with a delta interaction supported on a locally deformed curve, establishing spectral properties and conditions for the absence of discrete eigenvalues.

Contribution

It provides new insights into the spectral behavior of magnetic Schrödinger operators with delta interactions on deformed curves, extending known results to the magnetic case.

Findings

Essential spectrum remains unchanged by magnetic field.

Sufficient condition identified for the absence of discrete spectrum.

Discrete spectrum can be empty under certain geometric and magnetic conditions.

Abstract

We consider magnetic Schr\"odinger operator $H=(i \nabla +A)^2-\alpha \delta_\Gamma$ with an attractive singular interaction supported by a piecewise smooth curve $\Gamma$ being a local deformation of a straight line. The magnetic field $B$ is supposed to be nonzero and local. We show that the essential spectrum is $[-\frac14\alpha^2,\infty)$, as for the non-magnetic operator with a straight $\Gamma$, and demonstrate a sufficient condition for the discrete spectrum of $H$ to be empty.