Journal of the Mathematical Society of Japan

Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space

Hiroyuki MATSUMOTO and Naomasa UEKI

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Abstract

The spectra of the quadratic Hamiltonians on the twodimensional Euclidean space are determined completely by using the theory of the metaplectic representation. In some cases, the corresponding heat kernels are studied in connection with the well-definedness of the Wiener integrations. A proof of the Lévy formula for the stochastic area and a relation between the real and complex Hermite polynomials are given in our framework.

Article information

Source
J. Math. Soc. Japan, Volume 52, Number 2 (2000), 269-292.

Dates
First available in Project Euclid: 10 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213107373

Digital Object Identifier
doi:10.2969/jmsj/05220269

Mathematical Reviews number (MathSciNet)
MR1742802

Zentralblatt MATH identifier
0965.35101

Subjects
Primary: 35P05: General topics in linear spectral theory
Secondary: 35J10: Schrödinger operator [See also 35Pxx] 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 60H05: Stochastic integrals

Keywords
Spectrum quadratic Hamiltonians metaplectic representation heat kernels Wiener integrations L\'{e}vy formula Hermite polynomials

Citation

MATSUMOTO, Hiroyuki; UEKI, Naomasa. Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space. J. Math. Soc. Japan 52 (2000), no. 2, 269--292. doi:10.2969/jmsj/05220269. https://projecteuclid.org/euclid.jmsj/1213107373


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