Problem with the single-particle description and the spectra of intrinsic modes of degenerate boson-fermion systems

Abstract

The problem with the construction of the Gross-Pitaevskii (GP) equation and related wave functions in a medium is associated with the necessity to begin with a description of the configuration space and proceed to a description of the physical space. We show that the balance equations of the number of particles and momentum immediately follow from the multiparticle Schr\"odinger equation. From the obtained set of balance equations, the equation for the wave function in the medium coincides in form with the GP equation, where we can restrict ourselves only by the first-order terms of the interaction radius of the stress tensor of bosons, for dilute gases. As a generalization of the GP equation, we made allowance for the contribution of the third-order terms of the interaction radius to the stress tensor of bosons. For a system of particles that comprises an ultracold mixture of bosons and fermions, a two-kind quantum hydrodynamics is constructed for the third-order terms of the interaction radius. The spectrum of eigenmodes involves additional information on the interparticle interaction as a correction to the Bogoliubov spectrum.