Compactness in the ∂̄-Neumann problem, magnetic Schrödinger operators, and the Aharonov-Bohm effect

Michael Christ, Siqi Fu

Research output: Contribution to journalArticlepeer-review

23 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-40
Number of pages40
JournalAdvances in Mathematics
Volume197
Issue number1
DOIs
StatePublished - Oct 20 2005

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Aharonov-Bohm effect
  • Compactness
  • Hartogs domains
  • Kohn Laplacian
  • Magnetic Schrödinger operators
  • Property (P)
  • ∂̄-Neumann problem

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