New SUSYQM coherent states for PöschlTeller potentials: A detailed mathematical analysis

In a recent short note (Bergeron et al 2010 Europhys. Lett. 92 60003), we have presented the good properties of a new family of semi-classical states for PöschlTeller potentials. These states are built from a supersymmetric quantum mechanics (SUSYQM) approach and the parameters of these coherent states are points in the classical phase space. In this paper, we develop all the mathematical aspects that have been left out of the previous paper (proof of the resolution of unity, detailed calculations of the quantized version of classical observables and mathematical study of the resulting operators: problems of domains, self-adjointness or self-adjoint extensions). Some additional questions such as asymptotic behavior are also studied. Moreover, the framework is extended to a larger class of PöschlTeller potentials. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to Coherent states: mathematical and physical aspects.