Quantum B\"acklund Transformations: some ideas and examples

TL;DR

This paper explores the Hamiltonian interpretation of spectrality in Bäcklund transformations for integrable systems, linking it to separation of variables and quantum operators, with explicit examples and potential for explicit integration.

Contribution

It provides a mechanical interpretation of spectrality in BTs and connects quantum Baxter Q operators to Green's functions of the Schrödinger equation.

Findings

Spectrality property relates to Hamilton-Jacobi separation.

Explicit integration of models via BTs is possible.

Quantum Baxter Q operator can be interpreted as a propagator.

Abstract

In this work we give a mechanical (Hamiltonian) interpretation of the so called spectrality property introduced by Sklyanin and Kuznetsov in the context of B\"acklund transformations (BTs) for finite dimensional integrable systems. The property turns out to be deeply connected with the Hamilton-Jacobi separation of variables and can lead to the explicit integration of the underlying model through the expression of the BTs. Once such construction is given, it is shown, in a simple example, that it is possible to interpret the Baxter Q operator defining the quantum BTs us the Green's function, or propagator, of the time dependent Schr\"odinger equation for the interpolating Hamiltonian.