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Spherical Harmonics, the Weyl Transform and the Fourier Transform on the Heisenberg Group

Published online by Cambridge University Press:  20 November 2018

Daryl Geller
Daryl Geller*
Affiliation:
State University of New York at Stony Brook. Stony Brook, New York
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In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋa separable Hilbert space. Given certain (unbounded) operators W1,…,Wn,W1+, …, Wn+ on ℋ satisfying

(on a dense subspace D of ℋ) with all other commutators vanishing. Given also a function where ζ ∈ Cn. Let W = (W1 …, Wn) W+ = (W1+ …, Wn+). How does one associate to f an operator f(W, W+)? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W, Q = Im W+ which represent momentum and position. In this paper, however, we wish to exploit the unitary group on Cn and so prefer complex notation.)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 4 , 01 August 1984 , pp. 615 - 684
Copyright
Copyright © Canadian Mathematical Society 1984

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