Engliš, Miroslav; Peetre, Jaak On the correspondence principle for the quantized annulus. (English) Zbl 0865.30005 Math. Scand. 78, No. 2, 183-206 (1996). 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) Cited in 4 Documents 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 Keywords:quantized annulus; correspondence principle; Berezin quantization; reproducing kernel; weighted Bergman space; Berezin transform; asymptotic expansion; Laplace-Beltrami operator PDFBibTeX XML Full Text: EuDML