Spectral properties of Schrödinger operators on superconducting surfaces

Abstract

In this work we focus on the characterization of the space L^2 (S; \mathbb C) on Riemannian 2-manifolds S induced by a fixed magnetic vector potential A_0 in the nonlinear Ginzburg-Landau (GL) superconductivity model. The linear differential operator governing the GL model is the surface Schrödinger operator (i \nabla + A_0)^2 on S . We obtain a complete orthonormal system in L^2 (S; \mathbb C) from a collection of nontrivial solutions of the weak-form of the spectral problem associated with (i \nabla + A_0)^2 . Then, after proving that any member of this basis satisfies a higher regularity condition, we conclude that each is also an eigenfunction of the strong-form of the surface Schrödinger operator, and must satisfy a natural Neumann condition over any nonempty component of the manifold boundary \partial S . These results form the theoretical foundations used to develop efficient computational tools for simulating the Langevin version of the surface GL model.