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

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 Verdière and F. Truc. We also derive growth conditions for radially symmetric magnetic fields which lead to confinement of spin 1/2 particles.

18 pages; the main theorem has been expanded and generalized

References in corpus (2)

• Essential self-adjointness of Schroedinger type operators on manifolds
• On confining potentials and essential self-adjointness for Schrödinger operators on bounded domains in R^n

Cited by in corpus (1)

• Essential self-adjointness of symmetric first-order differential systems and confinement of Dirac particles on bounded domains in ${\mathbb{R}}^d$