# The quantum auxiliary linear problem & Darboux-Backlund transformations

## TL;DR

This paper develops a quantum analogue of the classical auxiliary linear problem and Darboux-Bäcklund transformations, providing new formulas for quantum Lax pairs and exploring their applications in open boundary conditions and quantum quenches.

## Contribution

It introduces a systematic construction of quantum Lax pair hierarchies and boundary reflection matrices, extending classical integrability concepts to quantum lattice models.

## Key findings

- Derived quantum time evolution operator formula for open systems
- Showed reflection K-matrix as a boundary Darboux-Bäcklund transformation
- Applied method to q-oscillator model illustrating quantum quench connections

## Abstract

We explore the notion of the quantum auxiliary linear problem and the associated problem of quantum Backlund transformations (BT). In this context we systematically construct the analogue of the classical formula that provides the whole hierarchy of the time components of Lax pairs at the quantum level for both closed and open integrable lattice models. The generic time evolution operator formula is particularly interesting and novel at the quantum level when dealing with systems with open boundary conditions. In the same frame we show that the reflection K-matrix can also be viewed as a particular type of BT, fixed at the boundaries of the system. The q-oscillator (q-boson) model, a variant of the Ablowitz-Ladik model, is then employed as a paradigm to illustrate the method. Particular emphasis is given to the time part of the quantum BT as possible connections and applications to the problem of quantum quenches as well as the time evolution of local quantum impurities are evident. A discussion on the use of Bethe states as well as coherent states and the path integral formulation for the study of the time evolution is also presented.