Eigenvalue Problem Of Schrodinger Operator Of Magnetic Field With Zeros

The eigenvalue problems of the magnetic Schrodinger operator in an applied magnetic field have been studied extensively in the pasta few years. In [LP3] and [PK] , estimate of the lowest eigenvalueof Schrodinger operator with a non-degenerately vanishing magnetic field in a bounded 2-dimensional domain is discussed and some estimations are involved. In this paper, we give an asymptotic estimate of the lowest eigenvalue for a strong or a weak magnetic field . In particular, we study several special cases of eigenvalue problems of the Schrodinger operator with a magnetic fields with zeros in a bounded 3-dimensional domain. We establish an asymptotic estimate of the lowest eigenvalue of a Schrodinger operator Of a magnetic fields with zeros of higher order in a bounded 2-dimensional domain. We will use this results for determining upper critical field of type 2 superconductors subjected to non-homogeneous applied magnetic fields.