Levy processes and Schroedinger equation

Details Levy processes and Schroedinger equation Levy processes and Schroedinger equation

2008-05-05

arxiv.org/abs/0805.0503v2

Physics

Condensed Matter

Statistical Mechanics

10 pages; changed the TeX documentclass; added references [21] and [22] and comments about them; changed definitions (11) and

Scientific paper

10.1016/j.physa.2008.11.035

We analyze the extension of the well known relation between Brownian motion and Schroedinger equation to the family of Levy processes. We consider a Levy-Schroedinger 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 Levy-Khintchin formula shows then how to write down this operator in an integro--differential form. When the underlying Levy process is stable we recover as a particular case the fractional Schroedinger equation. A few examples are finally given and we find that there are physically relevant models (such as a form of the relativistic Schroedinger equation) that are in the domain of the non-stable, Levy-Schroedinger equations.

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