In the early 1980's

Tokuda investigated the mean number of phonons in the cloud around the electron for both the optical and the piezoelectric polarons within the scheme of variational approach based on the unitary transformation and the method of the Lagrange multiplier. Many investigators studied the properties of the mean number of phonons of polaron by many other methods. Recently

the properties of the mean number of phonons of the polaron and the bound magnetopolaron in a symmetric quantum dot have been studied using the linear-combination operator and unitary transformation method by one of the present authors and co-worker. However

using linear-combination operator method

the properties of the mean number of phonons for polaron in an asymmetric quantum dot has not been investigated so far. In this paper

the influences the asymmetric parabolic confined potential on the properties of the mean number of phonons for the polaron in quantum dot are studied by using linear-combination operator and the unitary transformation method. The relations between the mean number of phonons and the vibrational frequency of the strong- and weak-coupling polaron in an asymmetric quantum dot with the transverse and longitudinal direction confinement strength of the quantum dot and the electron-phonon coupling strength are derived. Numerical calculations on the RbCl crystal

as an example

are performed

and the results show that the mean number of phonons and the vibrational frequency of the strong-coupling polaron will increase strongly with increasing the transverse and longitudinal confinement strength of the quantum dot. The mean number of phonons and the vibrational frequency of the strong-coupling polaron will increase with increasing the electron-phonon coupling strength.