Nodal Sets For Groundastates Of Magnetic Schr(?)dinger Operators

In this paper, we investigate nodal sets of groundstate eigenfunctions of magnetic Schrodinger operators acting on a non simply connected domain in R2. B.Helffer et al have investigated this problem with zero magnetic field, under the case in which the circulation of the magnetic vector potential lies in Z+1/2 (see [Hel]). In this paper, we applied the methods of H.Matano et al, which they have used when studying zeros of Ginzburg-Landau order parameter (see [EMQ]), to get a general characteristic of nodal sets of groundstate eigenfunctions of magnetic Schrodinger operators acting on a non simply connected domain in R2. Subsequently, under the condition of zero magnetic field, we generalize the technique of covering manifold, which has been used in [Hel], to the case of that the circulation of the mag-netic vector potential lies in Q. Based on this, we prove some original conclusions including that the nodal set consists of isolated points only in some cases, and the curves which belong to the nodal set could not slit the region in some other cases. Moreover, from these facts, we could understand that the property of nodal sets under appropriate conditions will completely differ from those discussed in [Hel].