Canonical transforms. IV. Hyperbolic transforms: Continuous series of SL(2,R) representations

Abstract

We consider the sl(2,R) Lie algebra of second-order differential operators given by the Schrödinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the Cεq continuous series of representations. The corresponding SL(2,R) Lie group is shown to be a group of integral transforms acting on a (two-component) space of square-integrable functions, with an integral (matrix) kernel involving Hankel and Macdonald functions. The subgroup bases for irreducible representations consist of Whittaker, power, Hankel, and Macdonald functions. We construct the operator which intertwines this realization of SL(2,R) with the more familiar Bargmann realization on functions on the unit circle. This operator implements the canonical transformation of the above Schrödinger systems to action and angle variables.