Nodal sets for groundstates of Schrodinger operators with zero magnetic field in non simply connected domains

We investigate nodal sets of magnetic Schrodinger operators with zero magne tic field, acting on a non simply connected domain in R-2. For the case of circulation 1/2 of the magnetic vector potential around each hole in the re gion, we obtain a characterisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole an d a fixed electric potential, we show that the first eigenvalue takes its h ighest value for circulation 1/2.