Gauge Invariant Eigenvalue Problems in ℝⁿ and in ℝⁿ₊

Abstract

This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane R 2 {\mathbb {R}}^{2} and in the half plane R + 2 {\mathbb {R}}^{2}_{+} . The estimates for the eigenvalues are obtained and the existence of the associate eigenfunctions is proved when c u r l A 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.