Path Integral Representation for Schroedinger Operators with Bernstein Functions of the Laplacian

TL;DR

This paper develops a path integral framework for generalized Schr"odinger operators using Bernstein functions of the Laplacian, extending Feynman-Kac formulas to include subordinated Brownian motion and applications to fractional and relativistic operators.

Contribution

It introduces a novel path integral representation for Schr"odinger operators with Bernstein functions, encompassing singular potentials and magnetic fields, and proposes a generalized Kato class.

Findings

Established path integral representations for generalized Schr"odinger operators.

Proved self-adjointness under singular conditions.

Derived hypercontractivity and comparison inequalities.

Abstract

Path integral representations for generalized Schr\"odinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with L\'evy subordinators is used, thereby the role of Brownian motion entering the standard Feynman-Kac formula is taken here by subordinated Brownian motion. As specific examples, fractional and relativistic Schr\"odinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which hypercontractivity of the associated generalized Schr\"odinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.