The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

TL;DR

This paper derives explicit formulas for Green's functions and transition probabilities for one-dimensional bosons and ASEP confined to the half-line, extending previous results to systems with boundary constraints.

Contribution

It provides the first explicit formulas for these systems with boundary confinement using a modified coordinate Bethe Ansatz.

Findings

Explicit Green's function for half-line Bose gas

Transition probability formula for half-line ASEP

Extension of Bethe Ansatz methods to boundary-constrained systems

Abstract

In this paper we find explicit formulas for: (1) Green's function for a system of one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.