On the spectrum of two-dimensional Schrödinger operators with spherically symmetric, radially periodic magnetic fields

We investigate the spectrum of the two-dimensional Schrödinger operator H = @ @x ia1(x; y) 2 @ @y ia2(x; y) 2 + V (x; y), where the magnetic field B(x; y) = @ @x a2 @ @y a1 and the electric potential V are spherically symmetric, i.e., B(x; y) = b(r), r = px2 + y2, and b is p-periodic, similarly for V . By considering two different gauges we get the following results: In case R p 0 b(s) ds = 0 the spectrum contains a semi-axis that consists alternately of intervals of absolutely continuous and dense point spectrum. In case R p 0 b(s) ds 6= 0 the essential spectrum is purely dense point spectrum and possibly there are spectral gaps.