A quantum waveguide with Aharonov Bohm magnetic field

TL;DR

This paper studies how an Aharonov-Bohm magnetic field affects the bound states of a quantum particle in a three-dimensional waveguide with specific boundary conditions, identifying conditions for the presence or absence of discrete spectrum.

Contribution

It introduces the effect of an Aharonov-Bohm magnetic field on the spectral properties of a quantum waveguide with mixed boundary conditions, providing critical thresholds for bound state existence.

Findings

Presence of a critical radius $a_0$ where discrete spectrum vanishes.

Magnetic field can suppress bound states below a certain threshold.

Conditions for the existence of bound states are established.

Abstract

In a previous study \cite{n} we investigate the bound states of the Hamiltonian describing a quantum particle living on three dimensional straight strip of width $d$. We impose the Neumann boundary condition on a disc window of radius $a$ and Dirichlet boundary conditions on the remained part of the boundary of the strip. We proved that such system exhibits discrete eigenvalues below the essential spectrum for any $a>0$. In the present work we study the effect of a magnetic filed of Aharonov-Bohm type when the magnetic field is turned on this system. Precisely we prove that in the presence of such magnetic filed there is some critical values of $a_0>0$, for which we have absence of the discrete spectrum for $\displaystyle 0<\frac{a}{d}<a_0$. We give a sufficient condition for the existence of discrete eigenvalues.