Lévy processes and Schrödinger equation

Nicola Cufaro Petroni and Modesto Pusterla

Physica A: Statistical Mechanics and its Applications, 2009, vol. 388, issue 6, 824-836

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.

Keywords: Stochastic mechanics; Lévy processes; Schrödinger equation (search for similar items in EconPapers)
Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:388:y:2009:i:6:p:824-836

DOI: 10.1016/j.physa.2008.11.035

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