Barut-girardello Type Of Coherent States For Qutumn System Of P（?）schl-teller Potential

It is well known that P?schl-Teller potential is widely used in quantum mechanics and molecular dynamics, it occupies a privileged place among the anharmonic oscillator potentials due to its applications in quantum mechanics and diatomic molecules. Therefore solving a quantum system associated with P?schl-Teller potential to get the energy eigenvalues and eigenstates have extraordinary significance. There are some reasonable and satisfying results which were got by different scholars with various methods. However the process is quite complicated, so it is keen for us to look for another easier way to solve Schr?dinger equations, with this way we can avoid solving second-order differential equations, as a result the problem is much easier, and we have been thinking about solving Schr?dinger equations associated with P?schl-Teller potential with algebraic methods for a long time. Although the SU(1,1) algebra method has been put forward for a long time, the coherent states about this alegbra has not been constructed yet. In this paper, we construct the coherent states of the Barut-Girardello coherent states(BG-CSs) type for the PT potentials, which have received less attention in the scientific literature. We obtain these CSs and demonstrate that they fulfil all conditions required by the coherent state. At the same time the Mandel parameter for the pure BG-CSs and Husimi’s and P-quasi distributions(for the mixed-thermal states) are also presented. Finally, the exponential form of the BG-CSs for the PT potential has been presented and enabled us to build Perelomov type CSs for the PT potential. Furthermore, we point out that the BG-CSs and the Perelomov type coherent states(PCSs) can be converted by Laplace transform.