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

Fumio Hiroshima, Takashi Ichinose, József Lrinczi

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    32 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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