Duality of two types ofSU(1, 1) coherent states and an intermediate type

Abstract

The two types of SU(1, 1) coherent states of Barut and Girardello and of Perelomov are dual in a sense that the operators in the eigenvalue equation and in the exponentials which create these types of coherent states from the lowest eigenstate (vacuum) form an asymmetric Heisenberg–Weyl algebra. A new type of SU(1, 1) coherent states which takes on an intermediate position between the two dual types of SU(1, 1) coherent states already mentioned is established and investigated. In this new type, the duality of the operators becomes a self-duality and the corresponding operators form a usual Heisenberg–Weyl algebra. Properties of the different SU(1, 1) coherent states are investigated for the realization of SU(1, 1) by one-dimensional quantum-mechanical potential problems leading to a quadratic law of energy-level spacing. Coherent SU(1, 1) phase states are discussed for this realization of SU(1, 1). It is shown that no wavepackets corresponding to any of the SU(1, 1) coherent states can exactly preserve their shape during time evolution.