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

Geller, Daryl

Canadian journal of mathematics, Tome 36 (1984) no. 4, pp. 615-684

Résumé

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 ∈ C n. Let W = (W1 ..., Wn) W+ = (W1+ ..., W n +). 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.)

Détail

DOI : 10.4153/CJM-1984-039-0

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