Gauge invariant eigenvalue problems in R-2 and in R-+(2)

This paper is devoted to the study of the eigenvalue problems for the Ginzb urg-Landau operator in the entire plane R-2 and in the half plane R-+(2). T he estimates for the eigenvalues are obtained and the existence of the asso ciate eigenfunctions is proved when curl A is a non-zero constant. These re sults are very useful for estimating the first eigenvalue of the Ginzburg-L andau operator with a gauge-invariant boundary condition in a bounded domai n, which is closely related to estimates of the upper critical field in the theory of superconductivity.