EXCITATION SPECTRUM OF DILUTE HE-3-HE-4 SOLUTION BY MODEL HAMILTONIAN

The Hamiltonian H of the He-3-He-4 mixture is considered to consist of the part of elementary excitations of He-3 and He-4, H(F) and H(B), and the interacting part between their density fluctuations, H(FB). We expand each density fluctuation in terms of the creation and annihilation of elementary excitations. By inserting this into H(FB) and diagonalizing H, we obtain the dispersion relation. Using sum rules, we solve the dispersion relation and obtain the energy spectrum for 5% He-3 concentration. The phonon-roton spectrum is decreased and the energy of the second branch almost unchanges. This result explains the decrease of a single roton energy observed in the fourth sound, oscillating disk and ion mobility experiments. We comment also on the result of the Raman scattering experiment.