TY - JOUR

T1 - Path integral representation for schrödinger operators with bernstein functions of the laplacian

AU - Hiroshima, Fumio

AU - Ichinose, Takashi

AU - Lrinczi, József

N1 - Funding Information:
F. H. acknowledges the support by JSPS Grant-in-Aid for Scientific Research (B) 20340032, JSPS Grant-in-Aid for Challenging Exploratory Research 22654018 and is thankful to Loughborough University, Paris XI University, and IHES, Bures-sur-Yvette, for hospitality. T. I. acknowledges support of JSPS Grant-in-Aid for Scientific Research (C) 20540161. J. L. thanks the Royal Society for an international travel grant, and the hospitality of Erwin Schrödinger Institute, Vienna, IHES, Bures-sur-Yvette, and Kyushu University, where various parts of this work have been done.

PY - 2012/7

Y1 - 2012/7

N2 - 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.

AB - 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.

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U2 - 10.1142/S0129055X12500134

DO - 10.1142/S0129055X12500134

M3 - Article

AN - SCOPUS:84862563972

VL - 24

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

SN - 0129-055X

IS - 6

M1 - 1250013

ER -