Curves of maximum modulus in coherent state representations

We consider functions of the form e-αz2 f(z), f entire analytic (Bargmann case), and yα f(z), f analytic in y>0 (Bergman case). We describe curves along which the moduli of such functions attain local maxima. Such a curve determines uniquely (up to a constant factor) a function f. If the curve is closed and if f has exactly one zero in the domain bounded by the curve, then the curve is a quantized circle. In this way we obtain the quantized classical orbits of the harmonic oscillator and the hydrogen atom, their wave functions and the correct spectra. In the Bargmann case the family of curves of maximum modulus is shown to consist of all straight lines and the quantized circles, while in the Bergman case this family is found to be much richer. It is believed that this represents an interesting approach to the study of some quantum mechanical operators