  We study the shift of the energy levels of electrons on helium surface due to
the coupling to the quantum field of surface vibrations. As in quantum
electrodynamics, the coupling is known, and it is known to lead to an
ultraviolet divergence of the level shifts. We show that there are diverging
terms of different nature and use the Bethe-type approach to show that they
cancel each other, to the leading-order. This resolves the long-standing
theoretical controversy and explains the existing experiments. The results
allow us to study the temperature dependence of the level shift. The
predictions are in good agreement with the experimental data.

Mesoscale and Nanoscale Physics