On the discrete spectrum of Schrödinger operators with strong magnetic fields of compact support

Rainer Hempel TU Braunschweig, Germany

We continue our analysis of the discrete spectrum of magnetic Schrödinger operators

H(λa⃗) = (−i∇ − λa⃗)^{2} + V (x), λ ≥ 0,

acting in L_{2}(R^{2}); here V is a bounded potential and a⃗ a magnetic vector potential of class C^{1} such that the associated magnetic field B = curl a⃗ has compact support \overline{Ω}. The interesting case is when the fluxes through the connected components of \overline{Ω} are non-zero.

We can now give a fairly complete description of the behavior of the discrete eigenvalues of H(λa⃗) at large coupling in the case where \overline{Ω} has a finite number of components. In the case of an infinite number of components, we need to make additional assumptions so far.