JournalsjstVol. 4, No. 3pp. 569–612

Spectral properties of Schrödinger operators on superconducting surfaces

  • Mahadevan Ganesh

    Colorado School of Mines, Golden, USA
  • Ty Thompson

    Colorado School of Mines, Golden, USA
Spectral properties of Schrödinger operators on superconducting surfaces cover

Abstract

In this work we focus on the characterization of the space L2(S;C)L^2 (S; \mathbb C) on Riemannian 2-manifolds SS induced by a fixed magnetic vector potential A0A_0 in the nonlinear Ginzburg-Landau (GL) superconductivity model. The linear differential operator governing the GL model is the surface Schrödinger operator (i+A0)2(i \nabla + A_0)^2 on SS. We obtain a complete orthonormal system in L2(S;C)L^2 (S; \mathbb C) from a collection of nontrivial solutions of the weak-form of the spectral problem associated with (i+A0)2(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 S\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