Gauge invariant eigenvalue problems in ℝ2 and in ℝ+ 2

Abstract

This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane ℝ2 and in the half plane R. The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when curl A is a non-zero constant. These results are very useful for estimating the first eigenvalue of the Ginzburg-Landau operator with a gauge-invariant boundary condition in a bounded domain, which is closely related to estimates of the upper critical field in the theory of superconductivity. © 1999 American Mathematical Society.