Vacancy excitation spectrum in solid He-4 and longitudinal phonons

The first microscopic ab initio calculation of the excitation spectrum of a vacancy in solid He-4 is reported. The energy-wave vector dispersion relation has been obtained at melting density within a development of the shadow wave function variational technique. The calculation of the excitation spectrum of a vacancy gives a bandwidth which ranges from 6 to 10 K in the hcp solid He-4, depending on the particular direction of the wave vector of the excitation. The effective mass of the vacancy turns out to be about 0.35 He-4 masses. We have also computed the spectrum of longitudinal phonons and we find rather good agreement with recent experimental results.