QUASI-COHERENT STATES AND THE SPECTRAL RESOLUTION OF THE Q-BOSE FIELD OPERATOR

The single mode q-Bose field for -1 less-than-or-equal-to q < 1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +/- (2/(1-q))1/2, thus becoming unbounded in the Bose limit (q --> 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatories. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q --> 1. Specifically it is shown that for 0 less-than-or-equal-to q < 1 the asymptotic growth is linear in time in contrast to the quadratic growth for q = 1.