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arXiv:1605.09273 · Journal of Physics A Mathematical and Theoretical

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

Maksim Tomchenko

Published Jan 6, 2017• 8 citations
arXivHTMLPDF

Abstract

Abstract 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 .
DOI
10.1088/1751-8121/aa5197

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Abstract We study the exact solutions for a one-dimensional system of N = 2; 3 spinless point bosons for zero boundary conditions. In this case, we are based on M Gaudin’s formulae obtained with the help of Bethe ansatz. We find the density profile ρ ( x ) and the nodal structure of a wave function for a set of the lowest states of the system for different values of the coupling constant γ ⩾ 0. The analysis shows that the ideal crystal corresponds to the quantum numbers (from Gaudin’s equations) n 1 = ⋯ = n N = N and to the coupling constant γ ≲ 1. We also find that the ground state (GS) of the system ( n 1 = ⋯ = n N = 1) corresponds to a liquid for any γ and any N ≫ 1. In this case, the wave function of the GS is nodeless, and the wave function of the ideal crystal has nodes.
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 Gaudin's solutions obtained with the help of the Bethe ansatz. We have also found the dispersion laws of the particle-like and hole-like excita tions under zero BCs. They coincide with the known dispersion laws obtained under periodic BCs.
We establish a remarkable relationship between the quantum Gaudin models with boundary and the classical many-body integrable systems of Calogero-Moser type associated with the root systems of classical Lie algebras (B, C and D). We show that under identification of spectra of the Gaudin Hamiltonians HjG with particles velocities q˙j of the classical model all integrals of motion of the latter take zero values. This is the generalization of the quantum-classical duality observed earlier for Gaudin models with periodic boundary conditions and Calogero-Moser models associated with the root system of the type A.
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 kkk in the dispersion law E(k)E(k)E(k) is unsuitable or the low-energy excitations are not phonons.