Dispersion Law for a One-Dimensional Weakly Interacting Bose Gas with Zero Boundary Conditions

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 Bogoliubov law \(E_{B}(k) = \sqrt{\left( \frac{\hbar ^{2} k^2}{2\,m}\right) ^{2} + n_{0}\nu (k)\frac{\hbar ^2 k^2}{m}}\) $E_B(k)=\sqrt{{\left(\frac{{ħ}^2{k}^2}{2m}\right)}^2+n_0\nu (k)\frac{{ħ}^2{k}^2}{m}}$ . 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.