In this work we focus on the characterization of the space on Riemannian 2-manifolds induced by a fixed magnetic vector potential in the nonlinear Ginzburg-Landau (GL) superconductivity model. The linear differential operator governing the GL model is the surface Schrödinger operator on . We obtain a complete orthonormal system in from a collection of nontrivial solutions of the weak-form of the spectral problem associated with . 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 . 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. 569–612DOI 10.4171/JST/79