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A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion

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Abstract

First we discuss the construction of self-adjoint Berezin–Toeplitz operators on weighted Bergman spaces via semibounded quadratic forms. To ensure semiboundedness, regularity conditions on the real-valued functions serving as symbols of these Berezin–Toeplitz operators are imposed. Then a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups is derived. All results are the consequence of a relation of Berezin–Toeplitz operators to Schrödinger operators defined via certain quadratic forms. The probabilistic expression is derived in conjunction with the Feynman–Kac–Itô formula.

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  1. Department of Physics, Princeton University, 337 Jadwin Hall, Princeton, NJ, 08544, U.S.A.

    Bernhard G. Bodmann

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Bodmann, B.G. A Construction of Berezin–Toeplitz Operators via Schrödinger Operators and the Probabilistic Representation of Berezin–Toeplitz Semigroups Based on Planar Brownian Motion. Mathematical Physics, Analysis and Geometry 5, 287–306 (2002). https://doi.org/10.1023/A:1020929223949

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