Quasimomentum of an elementary excitation for a system of point bosons under zero boundary conditions

Abstract

As is known, an elementary excitation of a many-particle system with boundaries is not characterized by a definite momentum. We obtain the formula for the quasimomentum of an elementary excitation for a one-dimensional system of N spinless point bosons under zero boundary conditions (BCs). In this case, we use the Gaudin's solutions obtained with the help of the Bethe ansatz. We have also found the dispersion laws of particle-like and hole-like excitations under zero BCs. They coincide with the known dispersion laws obtained for periodic BCs.

Low-Lying Energy Levels of a One-Dimensional Weakly Interacting Bose Gas under Zero Boundary Conditions

“…It is important that our solutions for E 0 and E(k) coincide with that ones obtained by the exactly solvable approach, based of the Bethe ansatz [7][8][9]. In addition, our solution for the density matrix F 1 (x 1 , x 2 )| T =0 coincides with the solution obtained within other methods for periodic BCs [39,[41][42][43][44][45][46][47][48].…”

“…This clearly shows that the Bogoliubov approximation is quite accurate for a finite 1D Bose system with weak coupling. Interestingly, for the point interaction, the Bogoliubov solutions E 0 and E(k) are approximately valid also at γ ∼ 1 even in the limit N → ∞ [5,8,9], which contradicts criterion (84). This means that the region of applicability of the Bogoliubov solutions is much wider than the region of applicability of the Bogoliubov method.…”

“…However, such influence is possible and, apparently, does not contradict any physical laws [6]. The solutions for a Bose system under zero BCs were found for a point interaction [7][8][9]. According to those results, c M.D.…”

“…. coincide with the corresponding levels [8,9] of a system of point bosons, which is described via the Bethe ansatz.…”

“…The solution obtained below is valid for a large number of particles: N > ∼ 1000. The system with N < ∼ 100 can be described in the exactly solvable approach [5,[7][8][9]23] (for a point-like potential) or by the multiconfigurational time-dependent Hartree method for bosons [24] (for a potential of the general form).…”