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arXiv · 0906.0103

Path Integral Representation for Schroedinger Operators with Bernstein Functions of the Laplacian

Fumio Hiroshima·Takashi Ichinose·Jozsef Lorinczi
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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 Feynman-Kac formula is taken here by subordinated 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 hypercontractivity of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

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Fumio Hiroshima, Takashi Ichinose, Jozsef Lorinczi. 2010-04-08. Path Integral Representation for Schroedinger Operators with Bernstein Functions of the Laplacian. https://arxiv.org/abs/0906.0103

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