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.