Riemann surface dynamics of periodic non-Hermitian Hamiltonians

Tobias Gulden, Michael Janas, Alex Kamenev

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2 Scopus citations


A number of physical problems, including statistical mechanics of 1D multivalent Coulomb gases, may be formulated in terms of non-Hermitian quantum mechanics. We use this example to develop a non-perturbative method of instanton calculus for non-Hermitian Hamiltonians. This can be seen as an extension of semiclassical methods in conventional quantum mechanics. Treating momentum and coordinate as complex variables yields a Riemann surface of constant complex energy. The classical and instanton actions are given by periods of this surface; we show how to obtain these via methods from algebraic topology. We demonstrate the accuracy of this analytic procedure in comparison with numerical simulations for a class of periodic non-Hermitian Hamiltonians, as well as the validity of the Bohr-Sommerfeld quantization and Gamow's formula in these cases.

Original languageEnglish (US)
Article number085001
JournalJournal of Physics A: Mathematical and Theoretical
Issue number8
StatePublished - Feb 28 2014


  • algebraic topology
  • ion channels
  • non-Hermitian quantum theory
  • statistical mechanics


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