Gauge Invariant Eigenvalue Problems in ${\mathbb{R}}^{2}$ and in ${\mathbb{R}}^{2}_{+}$

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.