Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains

Abstract

We establish an asymptotic estimate of the lowest eigenvalue μ ( b F ) \mu (b\mathbf {F}) of the Schrödinger operator − ∇ b F 2 -\nabla _{b\mathbf {F}}^{2} with a magnetic field in a bounded 2 2 -dimensional domain, where curl F \mathbf {F} vanishes non-degenerately, and b b is a large parameter. Our study is based on an analysis on an eigenvalue variation problem for the Sturm-Liouville problem. Using the estimate, we determine the value of the upper critical field for superconductors subject to non-homogeneous applied magnetic fields, and localize the nucleation of superconductivity.