Stability of the magnetic Schr\"odinger operator in a waveguide

TL;DR

This paper demonstrates that adding a magnetic field stabilizes the spectrum of the Schr"odinger operator in a waveguide against small geometric deformations and bending, contrasting with the known instability without magnetic fields.

Contribution

It introduces a magnetic Hardy-type inequality and proves spectral stability of the magnetic Schr"odinger operator under small local and bending deformations.

Findings

Magnetic field stabilizes the spectrum in waveguides.

Spectrum remains stable under small deformations and bending.

Established a magnetic Hardy inequality in the waveguide.

Abstract

The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schr\"odinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own.