Lévy processes and Schrödinger equation

We analyze the extension of the well known relation between Brownian motion and Schrödinger equation to the family of Lévy processes. We propose a Lévy– Schrödinger equation where the usual kinetic energy operator – the Laplacian – is generalized by means of a 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 recently proposed 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 possible Lévy–Schrödinger equations. PACS numbers: 02.50.Ey, 02.50.Ga, 05.40.Fb MSC numbers: 60G10, 60G51, 60J75