Three-dimensional magnetic Schr\"odinger operator with the potential supported in a tube

TL;DR

This paper investigates a 3D magnetic Schrödinger operator with a potential supported in a tubular region, demonstrating that the magnetic field does not alter the essential spectrum and providing conditions for the absence of discrete spectrum.

Contribution

The study introduces new methods to show magnetic fields do not affect the essential spectrum and establishes criteria for the discrete spectrum to be empty.

Findings

Magnetic field does not change the essential spectrum.

Conditions are provided for the discrete spectrum to be empty.

Potential supported in a tubular region along a deformed straight curve.

Abstract

In this paper, we study the following magnetic Schr\"odinger operator in $\mathbb{R}^3$: \[ H=(i \nabla +A)^2- \tilde{V}, \] where $\tilde{V}$ is non-negative potential supported over the tube built along a curve which is a local deformation of a straight one, and $B:=\mathrm{rot}(A)$ is a non-zero and local (i.e., a compact supported) magnetic field. Based on some new strategies, we first prove that the magnetic field does not change the essential spectrum of this system. Finally, in the last section of this paper, we establish the sufficient condition such that the discrete spectrum is empty.