Path integral representation for Schrödinger operators with Bernstein Functions of the Laplacian
journal contributionposted on 2016-07-11, 15:52 authored by Fumio Hiroshima, Takashi Ichinose, Jozsef Lorinczi
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.
- Mathematical Sciences
Published inREVIEWS IN MATHEMATICAL PHYSICS
Pages? - ? (40)
CitationHIROSHIMA, F., ICHINOSE, T. and LORINCZI, J., 2012. Path integral representation for Schrödinger operators with Bernstein Functions of the Laplacian. Reviews in Mathematical Physics, 24 (1250013), 40pp.
Publisher© World Scientific Publishing Co Pte Ltd
- AM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesElectronic version of an article published by World Scientific Publishing Company at: http://dx.doi.org/10.1142/S0129055X12500134