Gauge Invariant Eigenvalue Problems in ℝⁿ and in ℝⁿ₊

Lu, Kening; Pan, Xing-Bin

doi:10.1090/S0002-9947-99-02516-7

Gauge Invariant Eigenvalue Problems in $\mathbb {R}^n$ and in $\mathbb {R}^n_+$
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Trans. Amer. Math. Soc. 352 (2000), 1247-1276 Request permission

Abstract:

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.

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