Separation of Variables in the open XXX chain

TL;DR

This paper applies the Sklyanin separation of variables method to the open spin-1/2 XXX chain with non-diagonal boundary conditions, deriving the spectral problem and analyzing solution asymptotics.

Contribution

It extends the separation of variables approach to non-diagonal boundary conditions in the open XXX chain, connecting it to Bethe equations under boundary constraints.

Findings

Formulation of the spectral problem via a TQ-equation.

Derivation of Bethe equations for boundary parameters.

Analysis of asymptotic behavior of solutions.

Abstract

We apply the Sklyanin method of separation of variables to the reflection algebra underlying the open spin-1/2 XXX chain with non-diagonal boundary fields. The spectral problem can be formulated in terms of a TQ-equation which leads to the known Bethe equations for boundary parameters satisfying a constraint. For generic boundary parameters we study the asymptotic behaviour of the solutions of the TQ-equation.