Nature of Lieb's "hole" excitations and two-phonon states of a Bose gas

It is generally accepted that the hole'' and particle’’ excitations are two independent types of excitations of a one-dimensional system of point bosons. We show for a weak coupling that the Lieb’s ``hole’’ with the momentum $p=j2\pi \mathrm{/}L$ is $j$ identical interacting phonons with the momentum $2\pi \mathrm{/}L$ (here, $L$ is the size of the system, and $\mathrm{\hslash }=1$). We prove this assertion for $j=1,2$ by comparing solutions for a system of point bosons with solutions for a system of nonpoint bosons obtained in the limit of the point interaction. The additional arguments show that our conclusion should be true for any $j=1, 2, \ldots, N$. Thus, at a weak coupling, the holes are not a physically independent type of quasiparticles. Moreover, we find the solution for two interacting phonons in a Bose system with an interatomic potential of the general form at a weak coupling and any dimension (1, 2, or 3). It is also shown for a weak coupling that the largest number of phonons in a Bose system is equal to the number of atoms $N$. Finally, we have studied the structure of wave functions for the Tonks–Girardeau gas and found that the properties of quasiparticles in this regime are quite strange.