Path integral representation for schrödinger operators with bernstein functions of the laplacian

Fumio Hiroshima, Takashi Ichinose, József Lrinczi

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)

Abstract

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 FeynmanKac 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 L p-L q bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

Original languageEnglish
Article number1250013
JournalReviews in Mathematical Physics
Volume24
Issue number6
DOIs
Publication statusPublished - Jul 2012

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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