Intermediate-coupling exciton in a quantum well

The interaction expression of the surface-optical phonons with an electron (a hole) in a slab is extended to the case of a quantum well and the Hamiltonian H of the exciton-phonon system allowing for the Gabovish image potential is obtained for the first time. A double-time unitary transformation to the Hamiltonian H is carried out to obtain the Hamiltonian Hex of the exciton in a quantum well, and the motion of the exciton in the z direction is discussed in detail. The result obtained is suitable not only for a weak-coupling exciton system but also for an intermediate-coupling exciton system.