Correspondence Principle for the Quantized Annulus, Romanovski Polynomials, and Morse Potential

A general theory of quantization has been proposed by Berezin (see, e.g., his survey in Comm. Math. Phys.40 (1975), 153-174). In this paper we establish a weak form of the correspondence principle for the annulus, quantized according to Berezin. More precisely, we show that Bħ → I as ħ → 0, where I is the identity operator and Bħ the Berezin transform. We consider also spectral analysis on the annulus. In particular, we express the eigenfunctions of the relevant Laplacian in terms of certain Romanovski polynomials. Finally, we write down the expression for the analogue of the Morse potential in this case. We remark that similar considerations can be made, in principle, on any circular domain in the presence of a radial Hermitian metric of constant curvature.