Fumio Hiroshima, Takashi Ichinose and Jozsef Lorinczi
Path Integral Representation for Schr\"odinger Operators with 
Bernstein Functions of the Laplacian
(534K, pdf)

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, dia\-magnetic and energy comparison 
inequalities are also derived.