Schrödinger eigenbasis on a class of superconducting surfaces: ansatz, analysis, FEM approximations and computations (Q482403)

This paper is mainly concerned with the representation and computation of the eigenvalues and eigenfunctions of the surface Schrödinger operator, which governs a class of nonlinear Ginzburg-Landau superconductivity models on rotationally symmetric Riemannian 2-manifolds. In the first part, the authors develop a mathematical analysis of the weakly formulated surface Schrödinger spectral problem. The main result establishes that the members of a complete orthogonal system of eigenfunctions can be represented in a variable-separated parametric form. At the same time, an arbitrarily near spectral problem for the unknown factors in the representation is identified. In Section 3, the authors develop a numerical approach to compute, for any choice of the winding number, approximate eigenpairs of the surface Schrödinger operator through high-order finite element method approximations of the unknown factors in each separated form. Section 4 deals with the proof of the validity of this approximation computational framework for a class of smooth and nonsmooth surfaces with poles. Numerical applications illustrate several results of this paper.