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 self-adjoint, pseudo-differential 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.