Path Integrals for Relativistic Schrödinger Operators

Path integral methods for the investigation of the properties of Schrödinger operators go back to the pioneering works of Feynman and Kac. They have been successfully used for a long time but they have been restricted to nonrelativistic operators. We describe a class of stochastic processes, the so-called Levy processes (i.e. processes with stationary independent increments), and we explain how they can be used in the study of the relativistic operators and some of their abstract generalizations. We mimic the classical approach based on the use of Brownian motion and the Feynman-Kac’s formula. We describe the class of potentials which can be handled by this approach, we discuss the regularity properties of the semigroups, the decay of the eigenfunctions and the existence of bound states. We also discuss some of the open problems which are naturally unravelled.