2134/21957 Fumio Hiroshima Fumio Hiroshima Takashi Ichinose Takashi Ichinose Jozsef Lorinczi Jozsef Lorinczi Path integral representation for Schrödinger operators with Bernstein Functions of the Laplacian Loughborough University 2016 Feynman-Kac formula Generalized Schrodinger operator Bernstein function Levy process Spin Subordinator Poisson process Heat semigroup Mathematical Sciences not elsewhere classified 2016-07-11 15:52:10 Journal contribution https://repository.lboro.ac.uk/articles/journal_contribution/Path_integral_representation_for_Schr_dinger_operators_with_Bernstein_Functions_of_the_Laplacian/9385880 Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger 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 an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.