Geodesic
The two-branch spectrum of excitations of the Bose-condensate
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 13, Tome 224 (1995), pp. 310-317

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The two-branch spectrum of excitations of a weakly nonideal Bose-gas is obtained by using the integral over trajectories in the canonical ensemble with constraint. A rearrangment of the spectrum at the condensate density providing equilibrium of the attractive and repulsive forces in the interatomic potential is discovered. The influence of the quantum bosonic oscillations in the ground state on the stability of the branches is investigated. Bibliography: 5 titles. 
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     author = {V. S. Yarunin},
     title = {The two-branch spectrum of excitations of the {Bose-condensate}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {310--317},
     publisher = {mathdoc},
     volume = {224},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1995_224_a23/}
}
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V. S. Yarunin. The two-branch spectrum of excitations of the Bose-condensate. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 13, Tome 224 (1995), pp. 310-317. http://geodesic.mathdoc.fr/item/ZNSL_1995_224_a23/