Some remarks on the linearized operator about the radial solution for the Ginzburg-Landau equation

We consider the linearized operators, denoted L-d,L-1, of the Ginzburg-Landau operator Deltau + u(1 - \u\(2)) in R-2, about the radial solutions U-d,U-1(X) = f(d)(r)e(idtheta), for all d greater than or equal to 1. We state the correspondence between the real vector space of the bounded solutions of the equation L(d,1)w=0 and the eigenvalues of the linearized operators of the equations Deltau + 1/epsilon(2)u(1 - \u\(2)) = 0, in W B(0, 1), about the radial solutions u(d,epsilon)(x) = f(d)(r/epsilon)e(idtheta), that tend to 0 as epsilon tends to 0. (C) 2003 Published by Elsevier Ltd.