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

## TL;DR

This paper develops a semiclassical approach to analyze the energy spectrum of two interacting bosons in one dimension, providing explicit analytical results that match numerical and exact solutions, including bound states.

## Contribution

It extends semiclassical methods to a two-body problem with boundary conditions, offering analytical insights into the spectrum of interacting bosons in one dimension.

## Key findings

- Analytical expressions for the smooth density of states
- Excellent agreement with numerical calculations
- Accurate reproduction of exact energy levels

## 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.