Compactness in the ∂-neumann Problem, Magnetic Schrödinger Operators, and the Aharonov-bohm Effect

Compactness of the Neumann operator in the ∂̄-Neumann problem is studied for weakly pseudoconvex bounded Hartogs domains in two dimensions. A nonsmooth domain is constructed for which condition (P) fails to hold, yet the Kohn Laplacian still has compact resolvent. The main result, in contrast, is that for smoothly bounded Hartogs domains, the well-known sufficient condition (P) is equivalent to compactness. The analyses of compactness and condition (P) boil down to the asymptotic behavior of the lowest eigenvalues of two related sequences of Schrödinger operators, one with a magnetic field and one without, parametrized by a Fourier variable resulting from the Hartogs symmetry. The nonsmooth example is based on the Aharonov-Bohm phenomenon of quantum physics. For smooth domains not satisfying (P), we prove that there always exists an exceptional sequence of Fourier variables for which the Aharonov-Bohm effect is weak and thence that compactness fails to hold. This sequence can be very sparse, so that the lack of compactness is due to a rather subtle effect.