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Construction of Self-Adjoint Berezin-Toeplitz Operators on Kahler Manifolds and a Probabilistic Representation of the Associated Semigroups

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Bernhard G. Bodmann

Mathematical Physics · Submitted 19 Jul 2002

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim All results are the consequence of a relation between Berezin-Toeplitz operators and Schrodinger operators defined via certain quadratic forms.

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arXiv IDmath-ph/0207026 (version 2)
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CitationBernhard G. Bodmann. "Construction of Self-Adjoint Berezin-Toeplitz Operators on Kahler Manifolds and a Probabilistic Representation of the Associated Semigroups." arXiv:math-ph/0207026 (2002). https://pith.science/paper/HKL5QEFZ
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,551 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.14 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. As a first goal we develop self-adjointness criteria for Berezin-Toeplitz operators defined via quadratic forms. Then, following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. All results are the consequence of a relation between Berezin-Toeplitz operators and Schrodinger operators defined via certain quadratic forms. The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We investigate a class of operators resulting from a quantization scheme attributed to Berezin. contribution 0.61 0.71 0.03 asserted 0.96
s1 These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. background 0.99 0.10 0.03 asserted 1.00
s2 As a first goal we develop self-adjointness criteria for Berezin-Toeplitz operators defined via quadratic forms. contribution 0.98 0.86 0.08 asserted 0.97
s3 Then, following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. contribution 0.49 0.47 0.24 hedged possibility 1.00
s4 More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. method 0.83 0.56 0.08 asserted 1.00
s5 All results are the consequence of a relation between Berezin-Toeplitz operators and Schrodinger operators defined via certain quadratic forms. interpretation 0.61 0.80 0.48 asserted 1.00
s6 The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula. method 0.90 0.87 0.15 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. As a first goal we develop self-adjointness criteria for Berezin-Toeplitz operators defined via quadratic forms. Then, following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. All results are the consequence of a relation between Berezin-Toeplitz operators and Schrodinger operators defined via certain quadratic forms. The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula.

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