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Physical Review A

δ-function bosons in a one-dimensional potential well

You‐Quan Li

Published Jul 1, 1995• 7 citations
Publisher

Abstract

The quantum-mechanical problem of bosons, in a square well in one dimension and interacting with a \ensuremath{\delta} function, is analyzed and solved. The resulting Bethe-ansatz wave function is a sum, over the Weyl group of Lie algebra ${\mathit{B}}_{\mathit{N}}$, of nondiffractive waves. The secular equation of the spectrum is obtained and its thermodynamic limit is discussed.

DOI
10.1103/physreva.52.65

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The quantum-mechanical problem of bosons, in a square well in one dimension and interacting with a \ensuremath{\delta} function, is analyzed and solved. The resulting Bethe-ansatz wave function is a sum, over the Weyl group of Lie algebra BN{\mathit{B}}_{\mathit{N}}BN​, of nondiffractive waves. The secular equation of the spectrum is obtained and its thermodynamic limit is discussed.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.