Magnetic effects in curved quantum waveguides

TL;DR

This paper studies how magnetic fields and geometry influence the spectral properties of quantum waveguides, deriving effective models and asymptotic eigenvalue expansions as the cross section shrinks.

Contribution

It introduces new effective lower-dimensional models for magnetic quantum waveguides and provides asymptotic eigenvalue expansions considering magnetic and geometric effects.

Findings

Norm resolvent convergence to effective models

Asymptotic expansions for eigenvalues

Spectral stability via Hardy inequalities

Abstract

The interplay among the spectrum, geometry and magnetic field in tubular neighbourhoods of curves in Euclidean spaces is investigated in the limit when the cross section shrinks to a point. Proving a norm resolvent convergence, we derive effective, lower-dimensional models which depend on the intensity of the magnetic field and curvatures. The results are used to establish complete asymptotic expansions for eigenvalues. Spectral stability properties based on Hardy-type inequalities induced by magnetic fields are also analysed.