×

On the correspondence principle for the quantized annulus. (English) Zbl 0865.30005

Let \(\mathbb{A}_R= \{z: 1<|z|< R\}\) be an annulus in the complex plane, \(|dz|/ \omega (z)\) the Poincaré metric on \(\mathbb{A}_R\), and \(K_\alpha (x, \overline {y})\) the reproducing kernel for the weighted Bergman space \(A^2_\alpha (\mathbb{A}_R)\) consisting of analytic functions on \(\mathbb{A}_R\) square-integrable with respect to the measure \(\omega (z)^\alpha dE(z)\), where \(dE\) is the Lebesgue measure. The Berezin transform on \(\mathbb{A}_R\) is defined as \[ B_\alpha f(y)= \int_{\mathbb{A}_R} f(x) {{|K_\alpha (x, \overline {y} )|^2} \over {K_\alpha (y, \overline {y})}} \omega (x)^\alpha dE(x). \] The authors show that \[ \lim_{n\to \infty} {\textstyle {1\over 2n}} \omega (z)^{2n+ 2} K_{2n} (z,\overline {z})= {\textstyle {1\over \pi}} \] and derive from this the asymptotic expansion \[ B_\alpha f= f+ {\textstyle {1\over {\alpha+2}}} \widetilde {\Delta} f+ {\textstyle {1\over {(\alpha+ 2)^2}}} \bigl( {\textstyle {1\over 2}} \widetilde {\Delta}^2+ \widetilde {\Delta}\bigr) f+ O(\alpha^{-3}) \] which holds as \(\alpha\to+ \infty\) through the set of even integers. Here \(\widetilde {\Delta}= \omega (z)^2 \Delta\) is the Laplace-Beltrami operator corresponding to the Poincaré metric on \(\mathbb{A}_R\). Both assertions are shown to remain in force also in the limiting case of the punctured disc \(\mathbb{A}_0= \{z: 0< |z|< 1\}\). These results have immediate applications in the theory of quantization, as developed by Berezin, since they show that this method which has hitherto been known to work only on symmetric spaces can be used also on \(\mathbb{A}_R\) nd \(\mathbb{A}_0\).
Reviewer: M.Engliš (Praha)

MSC:

30C40 Kernel functions in one complex variable and applications
30E15 Asymptotic representations in the complex plane
81S99 General quantum mechanics and problems of quantization
Full Text: EuDML