Spectral properties of Schrödinger operators on superconducting surfaces

Abstract

In this work we focus on the characterization of the space L 2 .SI C/ on Riemannian 2-manifolds S induced by a xed magnetic vector potential A 0 in the nonlinear Ginzburg-Landau (GL) superconductivity model. e linear di erential operator governing the GL model is the surface Schrödinger operator .ir C A 0 / 2 on S. We obtain a complete orthonormal system in L 2 .SI C/ from a collection of nontrivial solutions of the weak-form of the spectral problem associated with .ir CA 0 / 2 . en, after proving that any member of this basis satis es 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.ese results form the theoretical foundations used to develop e cient computational tools for simulating the Langevin version of the surface GL model.

Schrödinger eigenbasis on a class of superconducting surfaces: Ansatz, analysis, FEM approximations and computations

“…Since C 0 = C , we take R 0 (s) = R(s) and Z 0 (s) = Z (s) for s ∈ I . We also observe that for > 0, each S lies in a nonpolar component in S (as defined in [11]), with the infima and suprema over I of both R and (R ) 2 s + (Z ) 2 s equal to positive constants.…”

“…The fact that there are positive constants which bound R from above and below on I allows us to apply Theorem 8 from Appendix B, then adapt the procedure detailed in Section 3 from [11] to obtain (15). We introduce the form…”

“…Whenever S has a boundary, we assume that each boundary point has a neighborhood in S whose intersection with S is simply connected, and we tag the natural Neumann boundary condition (i.e., (i∇ + Rθ)v ·ŝ = 0 on ∂ S , for v ∈ L 2 (∂ S; C), see [11]) to the domain of the Schrödinger operator. We shall assume that the functions R and Z in the parametrization x from (1) …”

“…In [11], we developed a robust theoretical framework to characterize the space L 2 (S; C), and found in particular that a complete orthonormal system of eigenfunctions of the Schrödinger operator exists. In this work, we extend previous results (through robust mathematical analysis) to concretely identify a variable-separated version of this eigenfunction system, then develop and verify (using parallel programming) an efficient algorithm for the numerical approximation of its members.…”

“…Following derivation in [11], the surface gradient formulation ∇u =ŝ u s R 2 s + Z 2 s +θ u θ R , u ∈ H 1 (S; C) (2) yields the operator parametrized representation …”