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 a Lagrange multiplier.The properties of the mean number of phonons of magnetopolaron are investigated by many investigators using many other methods.The properties of the mean number of phonons of the magnetopolaron in a surface of crystals and quantum wire have been studied using the linear-combination operator and unitary transformation method by one of the present authors and co-worker.Recently

the properties of the mean number of phonons of the magnetopolaron and the bound magneto-polaron in a symmetric quantum dot have been investigated using the linear-combination operator and the variational method of Pekar type method by one of the present authors and a co-worker.However

using a linear-combination operator method

the properties of the mean number of phonons for magnetopolaron in an asymmetric quantum dot have 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 magnetopolaron in quantum dot were studied by using a linear-combination operator and unitary transformation method.The relations between the mean number of phonons and the vibrational frequency of the strong-and weak-coupling magnetopolaron 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

were 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 direction and longitudinal direction confinement strength of the quantum dot

and will increase with increasing the electron-phonon coupling strength.