# Spectral properties of Schrödinger operators on superconducting surfaces

### Mahadevan Ganesh

Colorado School of Mines, Golden, USA### Ty Thompson

Colorado School of Mines, Golden, USA

## 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.

## Cite this article

Mahadevan Ganesh, Ty Thompson, Spectral properties of Schrödinger operators on superconducting surfaces. J. Spectr. Theory 4 (2014), no. 3, pp. 569–612

DOI 10.4171/JST/79