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 δ-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.

Partial Fermionization: Spectral Universality in 1D Repulsive Bose Gases

“…(B11) Setting ν = 0 one recovers the case involving only two particles A 1,1 = L λT √ 2 (−1 + e s erfc( √ s)) on a line [68], also related to a corresponding expression in fully balanced spin-one-half Fermi gases which has been derived in the context of second-order virial expansion [69].…”

From integrability to chaos in quantum Liouvillians

“…The BGS conjecture is well founded now in the semiclassical theory, valid for systems with a proper classical limit [9][10][11] and supported by overwhelming numerical and experimental evidence in many different quantum systems [12][13][14]. The situation in many-body quantum systems is much less clear, although some theoretical advances have been recently made [15][16][17]. Due to the symmetry under exchange of fermionic or bosonic particles, the classical limit cannot be properly defined.…”