Abstract
This article is a direct continuation of 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 KK-matrices describing the boundary conditions is triangular. In the present article we consider the most general integrable boundary conditions, namely the most general boundary KK-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 KK-matrix in a triangular form. In these settings the SoV resolution can be obtained along an extension of the method described in . 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 Baxterâs T-Q functional equation. We further describe scalar product properties of the separate states including eigenstates of the transfer matrix.
Highlights
Let us comment that the existence of the states ăΩÎČ | and |ΩÎČ ă can be proven by a general argument which we present in Appendix B
NQâ(λb|ÎČ )|ÎČ, h1, ..., hNă = |ÎČ, h1, ..., hNă Bh(λb) to prove that the vector in (188) coincides, up to the sign, with the vector (63) and so it is the corresponding transfer matrix eigenvector; one shows that the covector in (188) coincides with the covector (65). In this second article we have shown how to implement the Separation of Variables (SoV) method to characterize the transfer matrix spectrum for integrable models associated to the Bazhanov-Stroganov quantum Lax operator and to the most general integrable boundary conditions
For that purpose it was necessary to perform a gauge transformation so as to recast the problem in a form similar to the one studied in our first article, i.e., such that one of the boundary K-matrices becomes triangular after the gauge transformation
Summary
The out-of-equilibrium behavior of close and open physical systems has attracted a lot of interest motivated in particular by new experimental results, see e.g. [2,3,4,5,6,7,8,9,10]. [98, 111] and references therein, has to be adapted in a way similar to [80, 82] and generalized to these cyclic representations of the 6-vertex Yang-Baxter algebra Using this correspondence, the method and tools obtained in our first paper [1] can be used, leading to the complete characterization of the spectrum (again eigenvalues and eigenstates) of the general boundary transfer matrix. The most general boundary transfer matrix associated to the BazhanovStroganov Lax operator in the cyclic representations of the reflection algebra is defined by (λ) ⥠tra{Ka,+(λ) a,â(λ)} = a+(λ) â(λ) + d+(λ) â(λ) + b+(λ) â(λ) + c+(λ) â(λ) It is a one parameter family of commuting operators satisfying the following symmetries proprieties:.
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