Angle operators on weighted Bergman spaces

Abstract

Among the multiplication operators on weighted Bergman Hilbert spaces are those where the multiplying function (operator symbol) depends only on the angular polar coordinate in the unit disk: we call these “angle operators.” As these Hilbert spaces carry a CCR representation unitarily equivalent to the Schrödinger representation, angle operators are associated with quantum phase in the same way as are Toeplitz operators, for example. We determine the matrix elements of the angle operators with respect to the natural orthonormal basis on each of these spaces, and also with respect to the appropriate family of coherent states. By using a method of comparison with the corresponding results for Toeplitz operators, asymptotic expressions for the expectations and variances in these two families of states are obtained for the angle operators whose symbols are the polar angle function and its two complex exponentials. Notable is the fact that the asymptotic limit of the variance of the polar angle operator in the natural basis family is π2/3, which many authors take to be a requirement for a quantum phase operator.