Dispersion law for a one-dimensional weakly interacting Bose gas with zero boundary conditions

TL;DR

This paper derives the quasiparticle dispersion law for a one-dimensional weakly interacting Bose gas with zero boundary conditions, showing it matches the known Bogolyubov law for periodic boundaries, despite the increased analytical complexity.

Contribution

It provides the first derivation of the dispersion law for zero boundary conditions in a 1D Bose gas, extending the understanding beyond periodic boundary cases.

Findings

Dispersion law for zero BCs matches Bogolyubov law for periodic BCs.

Analysis of zero BCs is more complex than for periodic BCs.

The derived law confirms the universality of the Bogolyubov dispersion in 1D Bose gases.

Abstract

From the time-dependent Gross equation, we find the quasiparticle dispersion law for a one-dimensional weakly interacting Bose gas with a non-point interatomic potential and zero boundary conditions (BCs). The result coincides with the dispersion law for periodic BCs, i.e. the Bogolyubov law $E_{B}(k) = \sqrt{\left (\frac{\hbar^{2} k^2}{2m}\right )^{2} + n_{0}\nu(k)\frac{\hbar^2 k^2}{m}}$. In the case of periodic BCs, the dispersion law can be easily derived from Gross' equation. However, for zero BCs, the analysis is not so simple.