Levy processes and Schrodinger equation

We analyze the extension of the well known relation between Brownian motion and the Schrodinger equation to the family of the Levy processes. We consider a Levy-Schrodinger 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 Schrodinger equation. A few examples are finally given and we find that there are physically relevant models - such as a form of the relativistic Schrodinger equation - that are in the domain of the non stable Levy-Schrodinger equations. (C) 2008 Elsevier B.V. All rights reserved.