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Hilbert spaces of analytic functions and generalized coherent states. (English) Zbl 0941.81549
Generalized creation and annihilation operators are introduced, which in a certain limit \((q\rightarrow 1)\) become the usual boson creation and annihilation operators. These operators are bounded. From them generalized coherent states are constructed, which form an overcomplete basis in a Hilbert space \(H_q\) of analytic functions, and in the limit \(q\rightarrow 1\) become the usual coherent states of quantum optics. The scalar product in \(H_q\) is written in such a way that a “basic integration” is involved in its expression. It is shown that \(H_q\) reduces in the limit \(q\rightarrow 1\) to the Bargmann-Segal Hilbert space of entire functions and in the limit \(q\rightarrow 0\) to the Hardy-Lebesgue space.

81R30 Coherent states
46E20 Hilbert spaces of continuous, differentiable or analytic functions
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