Nonlocal field theory of quasiparticle scattering in dipolar Bose-Einstein condensates

We consider the propagation of quasiparticle excitations in a dipolar Bose-Einstein condensate, and derive nonlocal field theory scattering at stepwise inhomogeneity sound speed, obtained by tuning contact coupling part interaction on one side barrier. To solve this problem $ab$ $initio$, i.e., without prior assumptions form solutions, we reformulate Bogoliubov-de Gennes equation as singular integral equation. The latter is $novel$ $hypersingular$ type, having kernel which hypersingular only two isolated points. Deriving its solution, show that reveals continuum evanescent channels barrier absent for purely contact-interaction condensate. furthermore demonstrate performing discrete approximation kernel, achieves an excellent solution accuracy already moderate number discretization steps. Finally, non-monotonic nature system dispersion, corresponding to emergence roton minimum excitation spectrum, results peculiar features transmission reflection are nonexistent interactions.