In physics you may perceive coherent states (CS) either as a way of
description of lasers (etc) or, more generally, as one of ways of a
tentative conversion of a classical system into its possible quantum
descendants. In mathematics you may treat them either as one of
abstract off-springs of the algebra and group theory or as a broad
menu of concrete and friendly semi-classical-state structures. In
2010, in ref. [18], the present authors (plus J.-P. Gazeau), led by
the latter motivation, choose the specific Poeschl–Teller
interaction model (well known for its formal merits manifested, say,
in the so called supersymmetric quantum mechanics) and clarified its
CS-related phenomenological appeal (typically: the CS were
identified with points in the classical phase space). The present
text is, basically, a continuation. It offers a deeper mathematical
insight ranging from the proof of the resolution of unity and from
the explicit quantization of classical observables via some comments
on asymptotics up to the clarification of the details of the
operator domains and self-adjoint extensions.

MR2930523 Bergeron, H.; Siegl, P.; Youssef, A. New SUSYQM coherent
states for Pöschl-Teller potentials: a detailed mathematical
analysis. J. Phys. A 45 (2012), no. 24, 244028, 14 pp. 81Rxx