Polaron impurity states on a liquid helium film

We calculate the ground and the first excited states of an electron trapped within a dimple on a thin liquid helium film. The electron is localized by a positive impurity charge which is located on the substrate supporting the film. The mechanical equilibrium equation for the liquid helium surface is coupled to the Schrodinger equation which are solved using a variational approach. The energies of the ground and first excited states are obtained as a function of the helium film thickness and external electrical, and magnetic fields which are directly perpendicular to the two-dimensional system.