Collective excitations of a two-dimensional interacting Bose gas in anti-trap and linear external potentials

We present a method of finding approximate analytical solutions for the spectra and eigenvectors of collective modes in a two-dimensional system of interacting bosons subjected to a linear external potential or the potential of a special form $u(x,y)=\mu -u \cosh^2 x/l$, where $\mu$ is the chemical potential. The eigenvalue problem is solved analytically for an artificial model allowing the unbounded density of the particles. The spectra of collective modes are calculated numerically for the stripe, the rare density valley and the edge geometry and compared with the analytical results. It is shown that the energies of the modes localized at the rare density region and at the edge are well approximated by the analytical expressions. We discuss Bose-Einstein condensation (BEC) in the systems under investigations at $T\ne 0$ and find that in case of a finite number of the particles the regime of BEC can be realized, whereas the condensate disappears in the thermodynamic limit.