Quantum Magnetic Hamiltonians with Remarkable Spectral Properties

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