Quantum Magnetic Hamiltonians with Remarkable Spectral Properties

Abstract

The Hamiltonian, $H$, of a spinless particle moving in two dimensions in an axially symmetric magnetic field $B(\ensuremath{\rho})$ is considered. If $B(\ensuremath{\rho})\ensuremath{\sim}{\ensuremath{\rho}}^{\ensuremath{-}\ensuremath{\alpha}}$ for $\ensuremath{\rho}$, large with $0<\ensuremath{\alpha}<1$, then it is shown that $H$ has spectrum [$0, \ensuremath{\infty}$) with only eigenvectors and eigenvalues dense in [$0, \ensuremath{\infty}$). If $\ensuremath{\alpha}=1$, then the spectrum is a dense point spectrum in [0, $c$] for suitable $c$ and absolutely continuous in [$c, \ensuremath{\infty}$).

Locations

• Physical Review Letters - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF