In a recent short note [Bergeron H, Gazeau J P, Siegl P and Youssef A 2010View original: http://arxiv.org/abs/1112.0935
EPL 92 60003], we have presented the nice properties of a new family of
semi-classical states for P\"oschl-Teller potentials. These states are built
from a supersymmetric quantum mechanics approach and the parameters of these
"coherent" states are points in the classical phase space. In this article we
develop all the mathematical aspects that have been left apart in the previous
article (proof of the resolution of unity, detailed calculations of quantized
version of classical observables and mathematical study of the resulting
operators: problems of domains, self- adjointness or self-adjoint extensions).
Some additional questions as asymptotic behavior are also studied. Moreover,
the framework is extended to a larger class of P\"oschl-Teller potentials.