Semiclassics in a system without classical limit: The few-body spectrum of two interacting bosons in one dimension

Abstract

We present a semiclassical study of the spectrum of a few-body system consisting of two short-range interacting bosonic particles in one dimension, a particular case of a general class of integrable many-body systems where the energy spectrum is given by the solution of algebraic transcendental equations. By an exact mapping between $\delta$-potentials and boundary conditions on the few-body wave functions, we are able to extend previous semiclassical results for single-particle systems with mixed boundary conditions to the two-body problem. The semiclassical approach allows us to derive explicit analytical results for the smooth part of the two-body density of states that are in excellent agreement with numerical calculations. It further enables us to include the effect of bound states in the attractive case. Remarkably, for the particular case of two particles in one dimension, the discrete energy levels obtained through a requantization condition of the smooth density of states are essentially in perfect agreement with the exact ones.

10 pages, 8 figures, minor changes and corrections to match published verson in Physical Review E

References in corpus (4)

• Ultracold bosons with short-range interaction in regular optical lattices
• Excitation spectrum of bosons in a finite one-dimensional circular waveguide via the Bethe ansatz
• Controllable Asymmetric Matter-wave Beam Splitter and Ring Potential on an Atom Chip
• Exactly solvable interacting two-particle quantum graphs

Cited by in corpus (2)

• From integrability to chaos in quantum Liouvillians
• Partial Fermionization---Spectral Universality in 1D Repulsive Bose Gases