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Special Function Potentials for the Laplacian

Published online by Cambridge University Press:  20 November 2018

H. D. Fegan
H. D. Fegan*
Affiliation:
University of New Mexico, Albuquerque, New Mexico
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The purpose of this paper is to study the operator Δ + q. Here Δ is the Laplace–Beltrami operator on a compact Lie group G and q is a matrix coefficient of a representation of G. We are able to calculate the powers of Δ + q acting on the function qku. This is done in Section 2 and the reader is refered there for definitions of the special functions q and u.

The interest in the operator Δ + q comes originally from physics and in particular from the Schrödinger equation. This is described in [4]. Here we are restricting ourselves to mathematical questions and shall not consider any applications to physics.

In this paper we take the heat equation with potential as

(1.1)

with , the upper half plane, and initial data f(x, 0) = qk(x)u(x).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 5 , 01 October 1982 , pp. 1183 - 1194
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. J., Dieudonné, Special functions and linear representations of Lie groups, CBMS 1$ (Amer. Math. Soc., Providence, 1980)./Google Scholar
2. H. D. Fegan, , The spectrum of the Laplacian on forms over a Lie group, Pacific J. Math. 89 (1980)./Google Scholar
3. I. M., Gel'fand and Shilov, G. E., Generalized functions, vol. 1 (Academic Press, New York, 1964)./Google Scholar
4. Guillemin, V., Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977), 485–517./Google Scholar
5. Rudin, W., Functional analysis (McGraw-Hill, New York, 1973./Google Scholar
6. Weinstein, A., Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 4(1977), 883–892./Google Scholar
7. Weinstein, A., Eigenvalues of the Laplacian plus a potential, Proceedings of the International Congress of Mathematicians, Helsinki (1978), 803–805./Google Scholar