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Lévy processes and Schrödinger equation

Author

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  • Cufaro Petroni, Nicola
  • Pusterla, Modesto

Abstract

We analyze the extension of the well known relation between Brownian motion and the Schrödinger equation to the family of the Lévy processes. We consider a Lévy–Schrödinger equation where the usual kinetic energy operator–the Laplacian–is generalized by means of a selfadjoint, pseudodifferential operator whose symbol is the logarithmic characteristic of an infinitely divisible law. The Lévy–Khintchin formula shows then how to write down this operator in an integro-differential form. When the underlying Lévy process is stable we recover as a particular case the fractional Schrödinger equation. A few examples are finally given and we find that there are physically relevant models–such as a form of the relativistic Schrödinger equation–that are in the domain of the non stable Lévy–Schrödinger equations.

Suggested Citation

  • Cufaro Petroni, Nicola & Pusterla, Modesto, 2009. "Lévy processes and Schrödinger equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(6), pages 824-836.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:6:p:824-836
    DOI: 10.1016/j.physa.2008.11.035
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