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Excitation Spectrum of Dilute (3) He-(4) He Solution by Model Hamiltonian

The Hamiltonian H of the 3 He- 4 He mixture is considered to consist of the part of elementary excitations of 3 He and 4 He, 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% 3 He 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