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.