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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.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35P15 Estimates of eigenvalues in context of PDEs 82D55 Statistical mechanical studies of superconductors
##### Keywords:
Superconductivity; Ginzburg-Landau operator; eigenvalue
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##### References:
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