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(ρ) is considered. If B(ρ) ∼ ρ^(−α) for ρ, large with 0 < α < 1, then it is shown that H has spectrum [0, ∞) with only eigenvectors and eigenvalues dense in [0, ∞). If α = 1, then the spectrum is a dense point spectrum in [0, c] for suitable c and absolutely continuous in [c, ∞).