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Related Experiment Videos

The eigenfunction problem in higher dimensions: Asymptotic theory.

L Sirovich1, B W Knight

  • 1Brown University, Providence, RI 02912.

Proceedings of the National Academy of Sciences of the United States of America
|December 1, 1985
PubMed
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This study introduces a Wigner-Cotton transformation for spectral problems on infinite domains, enabling a multi-dimensional WKB theory. This provides practical tools for approximating spectra and eigenfunctions.

Area of Science:

  • Mathematical Physics
  • Quantum Mechanics
  • Spectral Theory

Background:

  • Spectral problems for linear operators on infinite domains are fundamental in various physics and mathematics fields.
  • Existing methods for analyzing these problems can be computationally intensive or limited in scope.

Purpose of the Study:

  • To develop a novel theoretical framework for analyzing spectral problems on fully infinite domains.
  • To extend the applicability of Wigner-type transformations and Wentzel-Kromers-Brillouin (WKB) theory to higher dimensions.
  • To provide practical computational tools for approximating spectra and eigenfunctions.

Main Methods:

  • Utilized a transformation technique, building upon Wigner's work, to analyze spectral properties.
  • Developed a multi-dimensional Wentzel-Kromers-Brillouin (WKB) theory for linear operators.

Related Experiment Videos

  • Applied the developed methods to several general examples for validation.

    • Main Results:

    • Established a rigorous WKB theory applicable to operators in more than one dimension.
    • Demonstrated the effectiveness of the Wigner-type transformation in deriving asymptotic spectral results.
    • Provided practical algorithms for the approximate evaluation of spectra and eigenfunctions.

    Conclusions:

    • The Wigner-type transformation offers a powerful approach to spectral problems on infinite domains.
    • The developed multi-dimensional WKB theory significantly enhances the analytical and computational capabilities in spectral analysis.
    • The methodology provides valuable tools for researchers in mathematical physics and quantum mechanics.