PATH INTEGRAL REPRESENTATION FOR SCHR¨ODINGER OPERATORS WITH BERNSTEIN FUNCTIONS OF THE LAPLACIAN

摘要

Path integral representations for generalized Schr¨odinger operators obtained under anclass of Bernstein functions of the Laplacian are established. The one-to-one correspondencenof Bernstein functions with L´evy subordinators is used, thereby the rolenof Brownian motion entering the standard Feynman–Kac formula is taken here by subordinatenBrownian motion. As specific examples, fractional and relativistic Schr¨odingernoperators with magnetic field and spin are covered. Results on self-adjointness of thesenoperators are obtained under conditions allowing for singular magnetic fields and singularnexternal potentials as well as arbitrary integer and half-integer spin values. Thisnapproach also allows to propose a notion of generalized Kato class for which an Lp-Lqnbound of the associated generalized Schr¨odinger semigroup is shown. As a consequence,ndiamagnetic and energy comparison inequalities are also derived