Unitarity of the SoV transform for the Toda chain

TL;DR

This paper proves the unitarity of the quantum separation of variables transform for the Toda chain using quantum inverse scattering methods, offering a new approach that can be adapted to other integrable models.

Contribution

It develops a technique to establish the unitarity of the SoV transform for the quantum Toda chain, applicable to other models with minimal modifications.

Findings

Proves unitarity of the SoV transform for the quantum Toda chain.

Uses quantum inverse scattering framework for the proof.

Provides an alternative to group theoretical proofs for integrable models.

Abstract

The quantum separation of variables method consists in mapping the original Hilbert space where a spectral problem is formulated onto one where the spectral problem takes a simpler "separated" form. In order to realise such a program, one should construct the map explicitly and then show that it is unitary. In the present paper, we develop a technique which allows one to prove the unitarity of this map in the case of the quantum Toda chain. Our proof solely builds on objects and relations naturally arising in the framework of the so-called quantum inverse scattering method. Hence, with minor modifications, it should be readily transposable to other quantum integrable models solvable by the quantum separation of variables method. As such, it provides an important alternative to the proof of the map's unitarity based on the group theoretical interpretation of the quantum Toda chain, which is absent for more complex quantum integrable models.