## Journal of the Mathematical Society of Japan

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

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

Hiroyuki MATSUMOTO and Naomasa UEKI

#### 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