Uniqueness of the solution of the Gaudin's equations, which describe a one-dimensional system of point bosons with zero boundary conditions

We show that the system of Gaudin's equations for quasimomenta k_{j}, which describes a one-dimensional system of spinless point bosons with zero boundary conditions, has the unique real solution for each set of quantum numbers n_{j}.Comment: 8 pages, v2: minor amendment

Calculation of the one-particle and two-particle condensates in He-II at T=0

We analyze the microstructure of He-II in the framework of the method of collective variables (CV), which was proposed by Bogolyubov and Zubarev and was developed later by Yukhnovskii and Vakarchuk. The logarithm of the ground-state wave function of He-II, ln(Psi_0), is calculated in the approximation of "two sums", i.e., as a Jastrow function and first (three-particle) correction. In the CV method equations for Psi_0 are deduced from the N-particle Schro'dinger equation. We also take into account the connection between the structure factor and Psi_0, which allows one to obtain Psi_0 from the structure factor of He-II, not from a model potential of interaction between He-II atoms. It should be emphasized that the model does not have any free parameters or functions. The amount of one-particle (N_1) and two-particle (N_2) condensates is calculated for the ground state of He-II: we find N_1=0.27N and N_2=0.53N in the Jastrow approximation for Psi_0, and, taking into account the three-particle correction to ln(Psi_0), we obtain N_1=0.06N (which agrees with the experiment) and $N_2=0.16N. In the approximation of "two sums", we also find that the higher s-particle condensates (s>2) are absent in He-II at T=0.Comment: 12 pages, 5 figure

Two dispersion curves for a one-dimensional interacting Bose gas in a vessel

In the framework of the Gross--Pitaevskii approach, we have considered the interacting Bose gas in a one-dimensional bounded domain and have found two different dispersion laws for phonons. One law coincides with the well-known Bogolyubov one (h\omega(k))^2 =(h^2 k^2/2m)^2 + qn\nu(k)h^2 k^2/m with q=1. The second law is new and is described by the same formula with q=1/2. The first solution corresponds to the single harmonic (as for the cyclic boundaries), and the second solution is represented by the set of harmonics forming a wave packet. Two solutions appear due to the interaction of harmonics in the integral of the Gross equation.Comment: 24 pages, 7 figures, v2, v3: some corrections in the tex

Point bosons in a one-dimensional box: the ground state, excitations and thermodynamics

We determine the ground-state energy and the effective dispersion law for a one-dimensional system of point bosons under zero boundary conditions. The ground-state energy is close to the value for a periodic system. But the dispersion law is essentially different from that for a periodic system, if the coupling is weak (weak interaction or high concentration) or intermediate. We propose also a new method for construction of the thermodynamics for a gas of point bosons. It turns out that the difference in the dispersion laws of systems with periodic and zero boundary conditions does not lead to a difference in the thermodynamic quantities. In addition, under zero boundary conditions, the microscopic sound velocity does not coincide with the macroscopic one. This means that either the method of determination of $k$ in the dispersion law $E(k)$ is unsuitable or the low-energy excitations are not phonons.Comment: 12 pages, 5 figures; v3: significant revision; the thermodynamics is added, the analysis of the $\delta$-function is corrected; accepted in J. Phys.

Possible critical regions for the ground state of a Bose gas in a spherical trap

With the help of perturbation theory, we study the ground state of a Bose gas in a spherical trap, using the solution in the Thomas--Fermi approximation as the zero approximation. We have found within a certain approximation that, in some very narrow intervals of values of the magnetic field of a trap, the solution deviates strongly from that in the Thomas--Fermi approximation. If the magnetic field is equal to one of such critical values, the size (or even the shape) of the condensate cloud should significantly differ from the Thomas--Fermi one.Comment: 7 pages, 3 figure

Thermodynamics of a one-dimensional system of point bosons: comparison of the traditional approach with a new one

We compare two approaches to the construction of the thermodynamics of a one-dimensional periodic system of spinless point bosons: the Yang--Yang approach and a new approach proposed by the author. In the latter, the elementary excitations are introduced so that there is only one type of excitations (as opposed to Lieb's approach with two types of excitations: particle-like and hole-like). At the weak coupling, these are the excitations of the Bogolyubov type. The equations for the thermodynamic quantities in these approaches are different, but their solutions coincide (this is shown below and is the main result). In this case, the new approach is simpler. An important point is that the thermodynamic formulae in the new approach for any values of parameters are formulae for an ensemble of quasiparticles with the Bose statistics, whereas a formulae in the traditional Yang--Yang approach have the Fermi-like one-particle form.Comment: 15 one-column pages, 6 figures; v2, v3: minor changes in the tex

Two dispersion curves for a one-dimensional interacting Bose gas under zero boundary conditions

The influence of boundaries and non-point character of interatomic interaction on the dispersion law has been studied for a uniform Bose gas in a one-dimensional vessel. The non-point character of interaction was taken into account using the Gross equation, which is more general than the Gross-Pitaevskii one. In the framework of this approach, the well-known Bogolyubov dispersion mode \hbar\omega(k)=[(\hbar^{2}k^{2}/2m) ^{2}+qn\nu(k)\hbar^{2}k^{2}/m]^{1/2} (q=1) was obtained, as well as a new one, which is described by the same formula, but with q= 1/2. The new mode emerges owing to the account of boundaries and the non-point character of interaction: this mode is absent when either the Gross equation for a cyclic system or the Gross-Pitaevskii equation for a cyclic system or a system with boundaries is solved. Capabilities for the new mode to be observed are discussed.Comment: 10 pages, 5 figure

On the strong influence of boundaries on the bulk microstructure of a uniform interacting Bose gas

It is usually assumed that the boundaries do not affect the bulk microstructure of an interacting uniform Bose gas. Therefore, the models use the most convenient cyclic boundary conditions. We show that, in reality, the boundaries affect strongly the bulk microstructure, by changing the ground-state energy E_0 and the energy of quasiparticles E(k). For the latter, we obtain the formula E^2 =(h^2 k^{2}/2m)^2 + 2^{-f}n\nu(k)(h^2 k^2/m) differing from the well-known Bogolyubov formula by the factor 2^{-f}, where f is the number of noncyclic coordinates. The Bogolyubov solution is also possible in the presence of boundaries, but it has a larger value of $E_{0}$ and should be unstable. The influence of boundaries is related to the topology.Comment: 6 pages, v.3: explanations are added; v.4: normalizing factors N^{-j} are regained in Eqs. (8), (9), (14

Bose crystal as a standing sound wave

A new class of solutions for Bose crystals with a simple cubic lattice consisting of N atoms is found. The wave function (WF) of the ground state takes the form \Psi_0=e^{S_{w}^{l}+S_{b}}*\prod_j [\sin{k_{l_x}x_{j}}\sin{k_{l_y}y_{j}}\sin{k_{l_z}z_{j}}], where e^{S_{b}} is the ground-state WF of a fluid, and \textbf{k}_l=(\pi/a_l, \pi/a_l, \pi/a_l) (a_l is the lattice constant). The state with a single longitudinal acoustic phonon is described by the WF \Psi_k=[\rho_{-k}+corrections + 7 permutations]\Psi_0, where the permutations give the terms with different signs of components of vector k. The structure of \Psi_k is such that the excitation corresponds, in fact, to the replacement of \textbf{k}_l in some triple of sines from \Psi_0 by \textbf{k}. Such a structure of \Psi_0 and \Psi_k means that the crystal is created by sound: the ground state of a cubic crystal is formed by N identical three-dimensional standing waves similar to a longitudinal sound. It is also shown that the crystal in the ground state has a condensate of atoms with \textbf{k}=\textbf{k}_l. The nonclassical inertia moment observed in crystals He-4 can be related to the synchronous tunneling of condensate atoms.Comment: 16 pages, 4 figure

On a fragmented condensate in a uniform Bose system

According to the well-known analysis by Nozi\'{e}res, the fragmentation of the condensate increases the energy of a uniform interacting Bose system. Therefore, at $T= 0$ the condensate should be nonfragmented. We perform a more detailed analysis and show that the result by Nozi\'{e}res is not general. We find that, in a dense Bose system, the formation of a crystal-like structure with a fragmented condensate is possible. The effect is related to a nonzero size of real atoms. Moreover, the wave functions studied by Nozi\'{e}res are not eigenfunctions of the Hamiltonian and, therefore, do not allow one to judge with confidence about the structure of the condensate in the ground state. We have constructed the wave functions in such a way that they are eigenfunctions of the Hamiltonian. The results show that the fragmentation of the condensate (quasicondensate) is possible for a finite one-dimensional uniform system at low temperatures and a weak coupling.Comment: 22 pages, 3 figures; v.2: Section 2 is extended, Figs. 2, 3 and Table 1 are adde