Scattering of neutrons on a Bose condensate

The spectrum of noncondensate excitations in neutron scattering on bosons i s obtained in the framework of the Bogoliubov models both for liquid He-4 a nd a dilute gas. The problem is solved using a path-integral representation of the partition function of the system. We describe the influence of scat tering of neutrons on a Bose condensate in a stationary (time-independent) picture in the Gibbs equilibrium ensemble. This influence is a stationary b oson response, and it depends on the initial neutron momentum k, transfer m omentum p, and the neutron-boson interaction lambda, which is related to th e scattering length. The contribution of the neutrons to the initial Bogoli ubov spectrum is found to be important for "quasi-elastic" scattering on th e noncondensate, while the contribution of deep inelastic scattering is sma ll; no contribution from elastic scattering on the Bose condensate is found . In the case of liquid Helium, the response is unlikely to be observable f or all values p. On the other hand, for a gas one may expect a visible effe ct, in particular for a small momentum transfer p and a small density of th e Bose condensate rho.