Threshold transition energies for Ginzburg-Landau functionals

In a previous paper (Almeida L 1996 Topological sectors for Ginzburg-Landau energies Rev. Mar. Iberoamericana to appear (preliminary version in author 's thesis, ENS Cachan, January 1996)) we studied the components of level se ts of Ginzburg-Landau energy functionals on multiply connected domains, and showed that they can be (partially) classified by an extended notion of to pological degree. We used this to show the existence of stable states and m ountain-pass solutions of Ginzburg-Landau equations. In this work, partly i nspired by the techniques we developed with Bethuel (Almeida L and Bethuel F 1998 Topological methods for the Ginzburg-Landau equation J. Math. Pures Appl. 77 1-49), we first improve the classification into topological sector s of our earlier mentioned paper, and then obtain quite precise estimates o n the threshold transition energies between different sectors. These enable us to, in the setting of the simple models considered, obtain the existenc e of states whose condensed wavefunction has a non-vanishing topological de gree and which are separated from the ground state by very high-energy barr iers-this phenomenon can be linked to the great stability of permanent curr ents in superconducting rings.