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.
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