Levy processes and Schroedinger equation
Abstract
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.
Cite
@article{arxiv.0805.0503,
title = {Levy processes and Schroedinger equation},
author = {Nicola Cufaro Petroni and Modesto Pusterla},
journal= {arXiv preprint arXiv:0805.0503},
year = {2014}
}
Comments
10 pages; changed the TeX documentclass; added references [21] and [22] and comments about them; changed definitions (11) and (12); added acknowledgments; small changes scattered in the text