Transfer matrix spectrum for cyclic representations of the 6-vertex reflection algebra II

TL;DR

This paper extends the spectral analysis of the transfer matrix for cyclic representations of the 6-vertex reflection algebra to the most general boundary conditions using the Separation of Variables method, providing a complete characterization of the spectrum.

Contribution

It introduces a gauge transformation approach to handle general boundary K-matrices and extends the SoV method to fully characterize the transfer matrix spectrum.

Findings

Complete spectral characterization via polynomial equations.

Solution of the spectral problem through an extended Baxter T-Q equation.

Analysis of scalar product properties of eigenstates.

Abstract

This article is a direct continuation of [1] where we begun the study of the transfer matrix spectral problem for the cyclic representations of the trigonometric 6-vertex reflection algebra associated to the Bazhanov-Stroganov Lax operator. There we addressed this problem for the case where one of the K-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary K-matrices satisfying the reflection equation. The spectral analysis is developed by implementing the method of Separation of Variables (SoV). We first design a suitable gauge transformation that enable us to put into correspondence the spectral problem for the most general boundary conditions with another one having one boundary K-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in [1]. The transfer matrix spectrum is then completely characterized in terms of the set of solutions to a discrete system of polynomial equations in a given class of functions and equivalently as the set of solutions to an analogue of Baxter's T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.