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A symbol calculus for Toeplitz operators.

C A Berger1, L A Coburn

  • 1Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY 10468.

Proceedings of the National Academy of Sciences of the United States of America
|May 1, 1986
PubMed
Summary

This study characterizes functions enabling Berezin-Toeplitz quantizations with symbol calculus. These functions exhibit limited oscillation at infinity, crucial for quantum mechanics and mathematical physics.

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Area of Science:

  • Mathematical Physics
  • Functional Analysis
  • Operator Theory

Background:

  • Berezin-Toeplitz quantization is a method for constructing quantum operators from classical functions.
  • Symbol calculus is essential for analyzing operator properties and their relationship to functions.
  • Compact operators play a significant role in the spectral theory of operators.

Purpose of the Study:

  • To fully characterize functions that allow for Berezin-Toeplitz quantizations with symbol calculus modulo compact operators.
  • To identify the precise conditions these functions must satisfy.
  • To advance the understanding of quantization techniques in mathematical physics.

Main Methods:

  • Analysis of function spaces on 2n-dimensional Euclidean space.

Related Experiment Videos

  • Investigation of properties of Berezin-Toeplitz operators.
  • Characterization of functions based on their behavior at infinity.

    • Main Results:

    • A complete characterization of the relevant functions is provided.
    • The key property identified is a condition of 'small oscillation at infinity'.
    • This condition precisely defines the functions for which the symbol calculus holds.

    Conclusions:

    • The study establishes a clear criterion for functions admitting symbol calculus in Berezin-Toeplitz quantization.
    • Understanding functions with 'small oscillation at infinity' is vital for developing and applying these quantization methods.
    • This work contributes to the theoretical framework of mathematical quantization and operator algebras.