Compactness in the d-bar Neumann problem, magnetic Schrodinger operators, and the Aharonov-Bohm effect

Abstract

Compactness of the d-bar Neumann operator is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth example is constructed in which condition (P) fails to hold, yet the Neumann operator is compact. The main result, in contrast, is that for smoothly bounded Hartogs domains, condition (P) of Catlin and Sibony is equivalent to compactness. The analyses of both compactness and condition (P) boil down to properties of the lowest eigenvalues of certain sequences of Schrodinger operators, with and without magnetic fields, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth counterexample is based on the Aharonov-Bohm phenomenon of quantum mechanics. For smooth domains, we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov-Bohm effect is quite weak. This sequence can be quite sparse, so that the failure of compactness is due to a rather subtle effect.