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Gauge invariant eigenvalue problems in \(\mathbb R^2\) and in \(\mathbb R^2_{+}\). (English) Zbl 1053.35124
Summary: This paper is devoted to the study of the eigenvalue problems for the Ginzburg-Landau operator in the entire plane \(\mathbb R^2\) and in the half plane \(\mathbb R^2_{+}\). 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.

35Q55 NLS equations (nonlinear Schrödinger equations)
35P15 Estimates of eigenvalues in context of PDEs
82D55 Statistical mechanical studies of superconductors
Full Text: DOI
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