Spectral theory for the q-Boson particle system

TL;DR

This paper develops a spectral theory for the q-Boson particle system, proving a Plancherel isomorphism, establishing Bethe ansatz completeness, and deriving moment formulas for related stochastic models.

Contribution

It introduces a spectral theory framework for the q-Boson system, including a Plancherel isomorphism and applications to stochastic heat equations and directed polymers.

Findings

Proves the completeness of the Bethe ansatz for the q-Boson generator.

Derives moment formulas for q-TASEP using Markov duality.

Establishes limits connecting q-Boson results to delta Bose gases and stochastic heat equations.

Abstract

We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation / Kardar-Parisi-Zhang equation.