Energy spectrum in the force field of an electrostatic image

A new model is suggested for describing localized electronic states in the vicinity of the surface of liquid helium. The model exploits nonuniqueness of self-adjoint extensions of the hamiltonian for an electron moving in an electrostatic image-induced potential near the flat interface isolator–vacuum. Unlike the Grimes model, the model proposed has only one phenomenological parameter, the mixing angle. It is shown that a proper choice of the mixing angle makes it possible to present the complete explanation of spectroscopic data on electrical dipole transitions between one-particle electronic states localized above the surface of liquid $^4\mathrm{He}$ or $^3\mathrm{He}$.