On essential self-adjointness for magnetic Schroedinger and Pauli operators on the unit disc in R^2

TL;DR

This paper establishes optimal growth conditions for magnetic fields near the boundary of the unit disk in R^2 that ensure quantum confinement of particles, addressing an open question and extending to spin 1/2 particles.

Contribution

It provides sharp, optimal growth criteria for magnetic fields to guarantee self-adjointness and confinement in magnetic Schrödinger and Pauli operators on the unit disk.

Findings

Optimal constants for magnetic field growth near boundary

Conditions ensuring confinement of spinless particles

Extension of results to spin 1/2 particles

Abstract

We study the question of magnetic confinement of quantum particles on the unit disk $\ID$ in $\IR^2$, i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $B(\vec x)$ near the boundary of the disk. In the spinless case we show that $B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}$, for $|\vec x|$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $\frac{\sqrt 3}{2}$ and $-\frac{1}{\sqrt 3}$ are optimal. This answers, in this context, an open question from Y. Colin de Verdi\`ere and F. Truc. We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.