Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions

TL;DR

This paper investigates the spectral properties of a magnetic Schrödinger operator in a waveguide with mixed boundary conditions, providing estimates on the window length for discrete spectrum absence and conditions for eigenvalues below the essential spectrum.

Contribution

It introduces new spectral estimates for the magnetic Schrödinger operator with combined boundary conditions in a waveguide, including bounds on window length and eigenvalue existence criteria.

Findings

Maximum window length for no discrete spectrum

Conditions for eigenvalues below the essential spectrum

Analysis includes both smooth and Aharonov-Bohm magnetic fields

Abstract

We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann boundary condition (see Section 2, Eq. (2.2) for the definition of this boundary condition}. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.