LPENSL - 2014\
IMB - 2014

[Open spin chains with generic integrable boundaries:\
Baxter equation and Bethe ansatz completeness from SOV]{}

[**N. Kitanine**,   **J.-M. Maillet**,   **G. Niccoli** ]{}

**Abstract**

Introduction
============

The functional characterization of the complete transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is a longstanding open problem. It has attracted much attention in the framework of quantum integrability producing so far only partial results. The interest in the solution of this problem is at least twofold. On the one hand, the quantum integrable system associated to the limit of the homogeneous chain, i.e. the open spin-1/2 XXZ quantum chain with arbitrary boundary magnetic fields, is an interesting physical quantum model. It appears, in particular, in the context of out-of-equilibrium physics ranging from the relaxation behavior of some classical stochastic processes, as the asymmetric simple exclusion processes [@EssD05; @EssD06], to the transport properties of the quantum spin systems [@SirPA09; @Pro11]. Their solution can lead to non-perturbative physical results and a complete and manageable functional characterization of their spectrum represents the first fundamental steps in this direction. On the other hand, it is important to remark that the analysis of the spectral problem of these integrable quantum models turned out to be quite involved by standard Bethe ansatz [@Bet31; @FadST79] techniques. Therefore, these quantum models are natural laboratories where to define alternative non-perturbative approach to their exact solution. Indeed, the algebraic Bethe ansatz, introduced for open systems by Sklyanin [@Skl88] based on the Cherednik’s reflection equation [@Che84], in the case of open XXZ quantum spin chains can be applied directly only in the case of parallel z-oriented boundary magnetic fields. Under these special boundary conditions the spectrum is naturally described by a finite system of Bethe ansatz equations. Moreover the dynamics of such systems can be studied by exact computation of correlation functions [@KitKMNST07; @KitKMNST08], derived from a generalisation of the method introduced in [@KitMT99; @KitMT00; @MaiT00] for periodic spin chains.

Introducing a Baxter $T$-$Q$ equation, Nepomechie [@Nep02; @Nep04] first succeeded to describe the spectrum of the XXZ spin chain with non-diagonal boundary terms in the case of an anisotropy parameter associated to the roots of unity; furthermore, the result was obtained there only if the boundary terms satisfied a very particular constraint relating the magnetic fields on the two boundaries. This last constraint was also used in [@CaoLSW03] to introduce a generalized algebraic Bethe ansatz approach to this problem inspired by papers of Baxter [@Bax72; @Bax72a] and of Faddeev and Takhtadjan [@FadT79] on the XYZ spin chain. This method has led to the first construction of the eigenstates of the XXZ spin chain with non z-oriented boundary magnetic fields and this construction has been obtained for a general anisotropy parameter, i.e., not restricted to the roots of unity cases[^1]. In [@YanZ07] a different version of this technique based on the vertex-IRF transformation was proposed but in fact it required one additional constraint on the boundary parameters to work. It is worth mentioning that even if these constrained boundary conditions are satisfied and generalized Bethe ansatz method gives a possibility to go beyond the spectrum, as it was done for the diagonal boundary conditions, no representation for the scalar product of Bethe vectors[^2] and hence for the correlation functions were obtained.

This spectral problem in the most general setting has then been also addressed by other approaches. It is worth mentioning a new functional method leading to nested Bethe ansatz equations presented in [@Gal08] for the eigenvalue characterization and analogous to those previously introduced in [@MurN05] by a generalized $T$-$Q$ formalism. The eigenstate construction has been considered in these general settings in [BasK05a,Bas06]{} by developing the so-called $q$-Onsager algebra formalism. In this last case the characterization of the spectrum is given by classifying the roots of some characteristic polynomials. More recently, in [@CaoYSW13-4] an ansatz $T$-$Q$ functional equations for the spin chains with non-diagonal boundaries has been proposed[^3].

It is extremely important to remark that in general all methods based on Bethe ansatz (or generalized Bethe ansatz) are lacking proofs of the completeness of the spectrum and in most cases the only evidences of completeness are based on numerical checks for short length chains. This is the case for the XXZ chain with non-diagonal boundary matrices with the boundary constraint for which the completeness of the spectrum description by the associated system of Bethe ansatz equations has been studied numerically [@Nep-R-2003; @Nep-R-2003add]. In the case of the XXZ chain with completely general non-diagonal boundary matrices some numerical analysis is also presented in [@CaoYSW13-4]. Further numerical analysis have been developed in a much simpler case of the isotropic XXX spin chain where the most general boundary conditions can be always reduced by using the $SU(2)$ symmetry to one diagonal and one non-diagonal boundary matrices. For the XXX chains the ansatz introduced in [@CaoYSW13-2] was also applied and the completeness of the Bethe ansatz spectrum was checked numerically [@CaoJYW2013]. It is also important to mention a simplified ansatz proposed by Nepomechie based on a standard second order difference functional $T$-$Q$ equation with an additional inhomogeneous term. The completeness of the Bethe ansatz spectrum has been verified numerically for small XXX chains in [@Nep-2013] while in [@Nep-W-2013] the problem of the description of some thermodynamical properties has been addressed.

These interesting developments attracted our attention in connection to the quantum separation of variables (SOV) method pioneered by Sklyanin [@Skl85; @Skl92]. The first analysis of the spin chain in the classical limit from this point of view was performed in [@Skl89a; @Skl89b]. This alternative approach allows to obtain (mainly by construction) the complete set of eigenvalues and eigenvectors of quantum integrable systems. In particular, it was recently developed [NicT10,N10-1,N10-2,GroMN12,GroN12,Nic12b,Nic13a,N13-1,Nic13b,Nic13c,Fald-KN13,FaldN13,GroMN13]{} for a large variety of quantum models not solvable by algebraic Bethe ansatz. Moreover it has been shown first in [@GroMN12] that once the SOV spectrum characterization is achieved manageable and rather universal determinant formulae can be derived for matrix elements of local operators between transfer matrix eigenstates. In particular, this SOV method was first developed in [@Nic12b] for the spin-1/2 representations of the 6-vertex reflection algebra with quite general non-diagonal boundaries and then generalized to the most general boundaries in [@Fald-KN13]. There, it gives the complete spectrum (eigenvalues and eigenstates) and already allows to compute matrix elements of some local operators within this most general boundary framework. However, it is important to remark that this SOV characterization of the spectrum is somehow unusual in comparison to more traditional characterizations like those obtained from Bethe ansatz techniques. More precisely, the spectrum is described not in terms of the set of solutions to a standard system of Bethe ansatz equations but is given in terms of sets of solutions to a characteristic system of $\mathsf{N}$ quadratic equations in $\mathsf{N}$ unknowns, $\mathsf{N}$ being the number of sites of the chain. While the clear advantage of this SOV characterization is that it permits to characterize completely the spectrum without introducing any ansatz one has to stress that the classification of the sets of solutions of the SOV system of quadratic equations represents a new problem in quantum integrability which requires a deeper and systematic analysis.

The aim of the present article is to show that the SOV analysis of the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebra on general inhomogeneous chains is strictly equivalent to a system of *generalized* Bethe ansatz equations. This ensures that this system of Bethe equations characterizes automatically the entire spectrum of the transfer matrix. More in detail, we prove that the SOV characterization is equivalent to a second order finite difference functional equation of Baxter type: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A} 
(-\lambda )Q(\lambda +\eta )+F(\lambda ),$$ which contains an inhomogeneous term $F(\lambda )$ independent on the $\tau $ and $Q$-functions and entirely fixed by the boundary parameters. It vanishes only for some special but yet interesting non-diagonal boundary conditions (corresponding to the boundary constraints mentioned above). One central requirement in our construction of this functional characterization is the polynomial character of the $Q$-function. Indeed, it is this requirement that allows then to show that a finite system of equations of generalized Bethe ansatz type can be used to describe the complete transfer matrix spectrum. Note that similar results on the reformulation of the SOV spectrum characterization in terms of functional $T$-$Q$ equations with $Q$-function solutions in a well defined model dependent set of polynomials were previously derived [@N10-1; @N10-2; @GroN12] for the cases of transfer matrices associated to cyclic representations of the Yang-Baxter algebra. The analysis presented here is also interesting as it introduces the main tools to generalize this type of reformulation to other classes of integrable quantum models. The article is organized as follows. In Section 2 we set the main notations and we recall the main results of previous papers on SOV necessary for our purposes. Section 3 contains the main results of the paper with the reformulation of the SOV characterization of the transfer matrix spectrum in terms of the inhomogeneous Baxter functional equation and the associated finite system of generalized Bethe ansatz equations. In Section 4 we define the boundary conditions for which the inhomogeneity in the Baxter equation identically vanishes, in this way deriving the completeness of standard Bethe ansatz equations. There, we moreover derive the SOV spectrum functional reformulation for the remaining boundary conditions compatibles with homogeneous Baxter equations. Section 5 contains the description of a set of discrete transformations which leave unchanged the SOV characterization of the spectrum in this way proving the isospectrality of the transformed transfer matrices. These symmetries are used to find equivalent functional equation characterizations of the spectrum which allow to generalize the results described in Section 3 and 4. In Section 6 we present the SOV characterization of the spectrum for the rational 6-vertex representation of the reflection algebra and the reformulation of the spectrum by inhomogeneous Baxter equation. Finally, in Section 7, we present a comparison with the known numerical results in the literature for both the XXZ and XXX chains; the evidenced compatibility suggests that even in the homogenous chains our spectrum description is still complete.

Separation of variable for spin-1/2 representations of the reflection algebra
=============================================================================

Spin-1/2 representations of the reflection algebra and open XXZ quantum chain
-----------------------------------------------------------------------------

The representation theory of the reflection algebra can be studied in terms of the solutions $\mathcal{U}(\lambda )$ (monodromy matrices) of the following reflection equation:$$R_{12}(\lambda -\mu )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda +\mu -\eta
)\,\mathcal{U}_{2}(\mu )=\mathcal{U}_{2}(\mu )\,R_{12}(\lambda +\mu -\eta )\,\mathcal{U}_{1}(\lambda )\,R_{21}(\lambda -\mu ). \label{bYB}$$Here we consider the reflection equation associated to the 6-vertex trigonometric $R$ matrix $$R_{12}(\lambda )=\left( 
\begin{array}{cccc}
\sinh (\lambda +\eta ) & 0 & 0 & 0 \\ 
0 & \sinh \lambda & \sinh \eta & 0 \\ 
0 & \sinh \eta & \sinh \lambda & 0 \\ 
0 & 0 & 0 & \sinh (\lambda +\eta )\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}),$$where $\mathcal{H}_{a}\simeq \mathbb{C}^{2}$ is a 2-dimensional linear space. The 6-vertex trigonometric $R$-matrix is a solution of the Yang-Baxter equation:$$R_{12}(\lambda -\mu )R_{13}(\lambda )R_{23}(\mu )=R_{23}(\mu )R_{13}(\lambda
)R_{12}(\lambda -\mu ).$$The most general scalar solution ($2\times 2$ matrix) of the reflection equation reads$$K(\lambda ;\zeta ,\kappa ,\tau )=\frac{1}{\sinh \zeta }\left( 
\begin{array}{cc}
\sinh (\lambda -\eta /2+\zeta ) & \kappa e^{\tau }\sinh (2\lambda -\eta ) \\ 
\kappa e^{-\tau }\sinh (2\lambda -\eta ) & \sinh (\zeta -\lambda +\eta /2)\end{array}\right) \in \text{End}(\mathcal{H}_{0}\simeq \mathbb{C}^{2}),  \label{ADMFKK}$$where $\zeta ,$ $\kappa $ and $\tau $ are arbitrary complex parameters. Using it and following [@Skl88] we can construct two classes of solutions to the reflection equation (\[bYB\]) in the 2$^{\mathsf{N}}$-dimensional representation space:$$\mathcal{H}=\otimes _{n=1}^{\mathsf{N}}\mathcal{H}_{n}.$$Indeed, starting from$$K_{-}(\lambda )=K(\lambda ;\zeta _{-},\kappa _{-},\tau _{-}),\text{ \ \ \ \ }K_{+}(\lambda )=K(\lambda +\eta ;\zeta _{+},\kappa _{+},\tau _{+}),$$where $\zeta _{\pm },\kappa _{\pm },\tau _{\pm }$ are the boundary parameters, the following boundary monodromy matrices can be introduced $$\begin{aligned}
\mathcal{U}_{-}(\lambda ) &=&M_{0}(\lambda )K_{-}(\lambda )\widehat{M}_{0}(\lambda )=\left( 
\begin{array}{cc}
\mathcal{A}_{-}(\lambda ) & \mathcal{B}_{-}(\lambda ) \\ 
\mathcal{C}_{-}(\lambda ) & \mathcal{D}_{-}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}), \\
\mathcal{U}_{+}^{t_{0}}(\lambda ) &=&M_{0}^{t_{0}}(\lambda
)K_{+}^{t_{0}}(\lambda )\widehat{M}_{0}^{t_{0}}(\lambda )=\left( 
\begin{array}{cc}
\mathcal{A}_{+}(\lambda ) & \mathcal{C}_{+}(\lambda ) \\ 
\mathcal{B}_{+}(\lambda ) & \mathcal{D}_{+}(\lambda )\end{array}\right) \in \text{End}(\mathcal{H}_{0}\otimes \mathcal{H}).\end{aligned}$$ These matrices $\mathcal{U}_{-}(\lambda )$ and $\mathcal{V}_{+}(\lambda )=\mathcal{U}_{+}^{t_{0}}(-\lambda )$ define two classes of solutions of the reflection equation (\[bYB\]). Here, we have used the notations:$$M_{0}(\lambda )=R_{0\mathsf{N}}(\lambda -\xi _{\mathsf{N}}-\eta /2)\dots
R_{01}(\lambda -\xi _{1}-\eta /2)=\left( 
\begin{array}{cc}
A(\lambda ) & B(\lambda ) \\ 
C(\lambda ) & D(\lambda )\end{array}\right)  \label{T}$$and $$\widehat{M}(\lambda )=(-1)^{\mathsf{N}}\,\sigma _{0}^{y}\,M^{t_{0}}(-\lambda
)\,\sigma _{0}^{y},  \label{Mhat}$$where $M_{0}(\lambda )\in $ End$(\mathcal{H}_{0}\otimes \mathcal{H})$ is the bulk inhomogeneous monodromy matrix (the $\xi _{j}$ are the arbitrary inhomogeneity parameters) satisfing the Yang-Baxter relation:$$R_{12}(\lambda -\mu )M_{1}(\lambda )M_{2}(\mu )=M_{2}(\mu )M_{1}(\lambda
)R_{12}(\lambda -\mu ).  \label{YB}$$The main interest of these boundary monodromy matrices is the property shown by Sklyanin [@Skl88] that the following family of transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\widehat{M}(\lambda )\}=\text{tr}_{0}\{K_{+}(\lambda )\mathcal{U}_{-}(\lambda )\}=\text{tr}_{0}\{K_{-}(\lambda )\mathcal{U}_{+}(\lambda
)\}\in \text{\thinspace End}(\mathcal{H}),  \label{transfer}$$defines a one parameter family of commuting operators in End$(\mathcal{H})$. The Hamiltonian of the open XXZ quantum spin 1/2 chain with the most general integrable boundary terms can be obtained in the homogeneous limit ($\xi _{m}=0$ for $m=1,\ldots ,\mathsf{N}$) from the following derivative of the transfer matrix (\[transfer\]):$$H=\frac{2(\sinh \eta )^{1-2\mathsf{N}}}{\text{tr}\{K_{+}(\eta /2)\}\,\text{tr}\{K_{-}(\eta /2)\}}\frac{d}{d\lambda }\mathcal{T}(\lambda )_{\,\vrule height13ptdepth1pt\>{\lambda =\eta /2}\!}+\text{constant,}  \label{Ht}$$and its explicit form reads: $$\begin{aligned}
H& =\sum_{i=1}^{\mathsf{N}-1}(\sigma _{i}^{x}\sigma _{i+1}^{x}+\sigma
_{i}^{y}\sigma _{i+1}^{y}+\cosh \eta \sigma _{i}^{z}\sigma _{i+1}^{z}) 
\notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{-}}\left[ \sigma _{1}^{z}\cosh \zeta
_{-}+2\kappa _{-}(\sigma _{1}^{x}\cosh \tau _{-}+i\sigma _{1}^{y}\sinh \tau
_{-})\right]  \notag \\
& +\frac{\sinh \eta }{\sinh \zeta _{+}}[(\sigma _{\mathsf{N}}^{z}\cosh \zeta
_{+}+2\kappa _{+}(\sigma _{\mathsf{N}}^{x}\cosh \tau _{+}+i\sigma _{\mathsf{N}}^{y}\sinh \tau _{+}).  \label{H-XXZ-Non-D}\end{aligned}$$Here $\sigma _{i}^{a}$ are local spin $1/2$ operators (Pauli matrices), $\Delta =\cosh \eta $ is the anisotropy parameter and the six complex boundary parameters $\zeta _{\pm }$, $\kappa _{\pm }$ and $\tau _{\pm }$ define the most general integrable magnetic interactions at the boundaries.

Some relevant properties
------------------------

The following quadratic linear combination of the generators $\mathcal{A}_{-}(\lambda ),$ $\mathcal{B}_{-}(\lambda ),$ $\mathcal{C}_{-}(\lambda )$ and $\mathcal{D}_{-}(\lambda )$ of the reflection algebra: $$\begin{aligned}
\frac{\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{\sinh (2\lambda -2\eta )}& =\mathcal{A}_{-}(\epsilon \lambda +\eta /2)\mathcal{A}_{-}(\eta /2-\epsilon
\lambda )+\mathcal{B}_{-}(\epsilon \lambda +\eta /2)\mathcal{C}_{-}(\eta
/2-\epsilon \lambda )  \label{q-detU_1} \\
& =\mathcal{D}_{-}(\epsilon \lambda +\eta /2)\mathcal{D}_{-}(\eta
/2-\epsilon \lambda )+\mathcal{C}_{-}(\epsilon \lambda +\eta /2)\mathcal{B}_{-}(\eta /2-\epsilon \lambda ),  \label{q-detU_2}\end{aligned}$$where $\epsilon =\pm 1$, is the *quantum determinant* . It was shown by Sklyanin that it is a central element of the reflection algebra$$\lbrack \mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ),\mathcal{U}_{-}(\mu )]=0.$$The quantum determinant plays a fundamental role in the characterization of the transfer matrix spectrum and it admits the following explicit expressions:$$\begin{aligned}
\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda ) &=&\mathrm{det}_{q}K_{-}(\lambda )\mathrm{det}_{q}M_{0}(\lambda )\mathrm{det}_{q}M_{0}(-\lambda )
\label{q-detU_-exp} \\
&=&\sinh (2\lambda -2\eta )\mathsf{A}_{-}(\lambda +\eta /2)\mathsf{A}_{-}(-\lambda +\eta /2),\end{aligned}$$where: $$\mathrm{det}_{q}M(\lambda )=a(\lambda +\eta /2)d(\lambda -\eta /2),
\label{bulk-q-det}$$is the bulk quantum determinant and$$\mathrm{det}_{q}K_{\pm }(\lambda )=\mp \sinh (2\lambda \pm 2\eta )g_{\pm
}(\lambda +\eta /2)g_{\pm }(-\lambda +\eta /2).$$Here, we used the following notations:$$\mathsf{A}_{-}(\lambda )=g_{-}(\lambda )a(\lambda )d(-\lambda ),\text{ \ }d(\lambda )=a(\lambda -\eta ),\text{ \ \ }a(\lambda )=\prod_{n=1}^{\mathsf{N}}\sinh (\lambda -\xi _{n}+\eta /2),  \label{eigenA}$$and$$g_{\pm }(\lambda )=\frac{\sinh (\lambda +\alpha _{\pm }-\eta /2)\cosh
(\lambda \mp \beta _{\pm }-\eta /2)}{\sinh \alpha _{\pm }\cosh \beta _{\pm }},  \label{g_PM}$$where $\alpha _{\pm }$ and $\beta _{\pm }$ are defined in terms of the boundary parameters by:$$\sinh \alpha _{\pm }\cosh \beta _{\pm }=\frac{\sinh \zeta _{\pm }}{2\kappa
_{\pm }},\text{ \ \ \ \ \ }\cosh \alpha _{\pm }\sinh \beta _{\pm }=\frac{\cosh \zeta _{\pm }}{2\kappa _{\pm }}.  \label{alfa-beta}$$

\[normality\]The transfer matrix $\mathcal{T}(\lambda )$ is an even function of the spectral parameter $\lambda $:$$\mathcal{T}(-\lambda )=\mathcal{T}(\lambda ),  \label{even-transfer}$$and it is central for the following special values of the spectral parameter: $$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\lambda (\mathsf{N}+2)}\mathcal{T}(\lambda ) &=&2^{-(2\mathsf{N}+1)}\frac{\kappa _{+}\kappa
_{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}},
\label{Central-asymp} \\
\mathcal{T}(\pm \eta /2) &=&(-1)^{\mathsf{N}}2\cosh \eta \mathrm{det}_{q}M(0),
\label{Central-1} \\
\mathcal{T}(\pm (\eta /2-i\pi /2)) &=&-2\cosh \eta \coth \zeta _{-}\coth
\zeta _{+}\mathrm{det}_{q}M(i\pi /2).  \label{Central-2}\end{aligned}$$Moreover, the monodromy matrix $\mathcal{U}_{\pm }(\lambda )$ satisfy the following transformation properties under Hermitian conjugation:

-   Under the condition $\eta \in i\mathbb{R}$ (massless regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(-\lambda
    ^{\ast })\right] ^{t_{0}},  \label{ml-Hermitian_U}$$ for $\{i\tau _{\pm },i\kappa _{\pm },i\zeta _{\pm },\xi
    _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$

-   Under the condition $\eta \in \mathbb{R}$ (massive regime), it holds: $$\mathcal{U}_{\pm }(\lambda )^{\dagger }=\left[ \mathcal{U}_{\pm }(\lambda
    ^{\ast })\right] ^{t_{0}},  \label{m-Hermitian_U}$$for $\{\tau _{\pm },\kappa _{\pm },\zeta _{\pm },i\xi
    _{1},...,i\xi _{\mathsf{N}}\}\in \mathbb{R}^{\mathsf{N}+3}.$

So under the same conditions on the parameters of the representation it holds: $$\mathcal{T}(\lambda )^{\dagger }=\mathcal{T}(\lambda ^{\ast }),
\label{I-Hermitian_T}$$i.e. $\mathcal{T}(\lambda )$ defines a one-parameter family of normal operators which are self-adjoint both for $\lambda $ real and purely imaginary.

SOV representations for $\mathcal{T}(\protect\lambda )$-spectral problem
------------------------------------------------------------------------

Let us recall here the characterization obtained in [@Nic12b; @Fald-KN13] by SOV method of the spectrum of the transfer matrix $\mathcal{T}(\lambda )$. First we introduce the following notations:$$X_{k,m}^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \left(
-1\right) ^{i}\left( 1-r\right) \eta +\tau _{-}-\tau_{+}+(-1)^{k}(\alpha
_{-}+\beta _{-})-(-1)^{m}(\alpha _{+}-\beta _{+})+i\pi (k+m),
\label{SOV-cond-}$$and by using these linear combinations of the boundary parameters we introduce the set $N_{SOV}\subset\mathbb{C}^6$ of boundary parameters for which the separation of variables cannot be applied directly. More precisely $$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha_{-},\beta _{-})\in N_{SOV},$$ if $\exists (k,h,m,n)\in \left\{ 0,1\right\} $ such that $$X_{k,m}^{(0,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0 \quad\text{and}\quad
X_{h,n}^{(1,\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0.$$All the results in the following will be obtained for the generic values of the boundary parameters, not belonging to this set. The SOV method applicability can be further extended applying the discrete symmetries discussed in the Section \[sect-descretesym\].

Following [@Fald-KN13] we define the functions:$$\begin{aligned}
g_{a}(\lambda ) &=&\frac{\cosh ^{2}2\lambda -\cosh ^{2}\eta }{\cosh
^{2}2\zeta _{a}^{(0)}-\cosh ^{2}\eta }\,\prod_{\substack{ b=1 \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}\quad \text{ \ for }a\in \{1,...,\mathsf{N}\}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{\sinh (2\lambda +\eta )}{\sinh 2\lambda }g_{+}(\lambda )g_{-}(\lambda )a(\lambda )d(-\lambda ),\end{aligned}$$and$$\begin{aligned}
f(\lambda )=& \frac{\cosh 2\lambda +\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh \eta -\cosh
2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2)  \notag \\
& -(-1)^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh \eta }{2\cosh \eta }\prod_{b=1}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
\eta +\cosh 2\zeta _{b}^{(0)}}\mathbf{A}(\eta /2+i\pi /2)  \notag \\
& +2^{(1-\mathsf{N})}\frac{\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})}{\sinh \zeta _{+}\sinh \zeta _{-}}(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)\prod_{b=1}^{\mathsf{N}}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}),
\label{f-function}\end{aligned}$$where$$\zeta _{n}^{(h_{n})}=\xi _{n}+(h_{n}-\frac{1}{2})\eta \quad \forall n\in
\{1,...,\mathsf{N}\},\text{ }h_{n}\in \{0,1\}\text{.}$$We can now recall the main result on the characterization of the set $\Sigma
_{\mathcal{T}}$ formed by all the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$.

\[C:T-eigenstates-\]Let $(\tau _{+},\alpha _{+},\beta _{+},\tau
_{-},\alpha _{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic:$$\xi _{a}\neq \pm\xi _{b}+r\eta\text{\ \ mod\,}2\pi \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\},  \label{xi-conditions}$$then $\mathcal{T}(\lambda )$ has simple spectrum and the set of its eigenvalues $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,
\label{Interpolation-Form-T}$$where $\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\sinh (\eta +2\xi _{n})\sinh (\eta
-2\xi _{n})},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},
\label{Quadratic System}$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.

Inhomogeneous Baxter equation 
==============================

Here we show that the SOV characterization of the spectrum admits an equivalent formulation in terms of a second order functional difference equation of Baxter type:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ),  \label{Inhom-Baxter-Eq}$$which contains a non-zero inhomogeneous term $F(\lambda )$ non-zero for generic integrable boundary conditions and the $Q$-functions are [*trigonometric polynomials*]{}. In this paper we will call $f(\lambda)$ a trigonometric polynomial of degree $\mathsf{M}$ if $e^{\mathsf{M}\lambda}\,f(\lambda)$ is a polynomial of $e^{2\lambda}$ of degree $\mathsf{M}$. Most trigonometric polynomials we will consider in the following sections will be even functions of $\lambda$ and will satisfy an additional condition $f({\lambda}+i\pi)=f({\lambda})$. It is easy to see in this situation that such functions can be written as polynomials of $\cosh 2\lambda$.

Main functions in the functional equation
-----------------------------------------

Let $Q(\lambda )$ be an even trigonometric polynomial of degree $2\mathsf{N}$. It can be written in the following form:$$\begin{aligned}
Q(\lambda )& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1  \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh 2\zeta
_{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\zeta
_{a}^{(0)}\right)  \label{Q-form1} \\
& =2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) ,  \label{Q-form2}\end{aligned}$$where from now on the $Q(\zeta _{a}^{(0)})$ are arbitrary complex numbers or similarly the $\lambda _{a}$ are arbitrary complex numbers. Then, introducing the function:$$Z_{Q}(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta )$$we can prove the following Lemma

Let $Q(\lambda )$ be any function of the form $\left( \ref{Q-form2}\right) $ then the associated function $Z_{Q}(\lambda )$ is an even trigonometric polynomial of degree $4\mathsf{N}+4$ of the following form:$$Z_{Q}(\lambda )=\sum_{a=0}^{2(\mathsf{N}+1)}z_{a}\cosh ^{a}2\lambda ,\text{
with }z_{2(\mathsf{N}+1)}=\frac{2\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{\sinh \zeta
_{+}\sinh \zeta _{-}}.$$

The fact that the function $Z_{Q}(\lambda )$ is even in $\lambda $ is a trivial consequence of the fact that $Q(\lambda )$ is even; in fact, it holds:$$\begin{aligned}
Z_{Q}(-\lambda ) &=&\mathbf{A}(-\lambda )Q(-\lambda -\eta )+\mathbf{A}(\lambda )Q(-\lambda +\eta )  \notag \\
&=&\mathbf{A}(-\lambda )Q(\lambda +\eta )+\mathbf{A}(\lambda )Q(\lambda
-\eta )=Z_{Q}(\lambda ).\end{aligned}$$The fact that $Z_{Q}(\lambda )$ is indeed a trigonometric polynomial follows from its definition once we observe that $\lambda=0$ is not a singular point and the following identity holds:$$\lim_{\lambda \rightarrow 0}Z_{Q}(\lambda )=2g_{+}(0)g_{-}(0)a(0)a(-\eta
)Q(0)\cosh \eta .$$Now the functional form of $Z_{Q}(\lambda )$ is a consequence of the following identities:$$Z_{Q}(\lambda +i\pi )=Z_{Q}(\lambda ),\text{ \ }\lim_{\lambda \rightarrow
\pm \infty }\frac{Z_{Q}(\lambda )}{e^{\pm 4(\mathsf{N}+1)\lambda }}=\frac{\kappa _{+}\kappa _{-}\cosh (\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )}{2^{(2\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}},$$where the second identity follows from:$$\begin{aligned}
\lim_{\lambda \rightarrow \pm \infty }e^{\mp (2\mathsf{N}+4)\lambda }\mathbf{A}(\lambda )& =2^{-2(\mathsf{N}+1)}\frac{\kappa _{+}\kappa _{-}\exp \pm
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1)\eta )}{\sinh
\zeta _{+}\sinh \zeta _{-}}, \\
\lim_{\lambda \rightarrow \pm \infty }e^{\mp 2\mathsf{N}\lambda }Q(\lambda
)& =1.\end{aligned}$$

On the need of an inhomogeneous term in the functional equation
---------------------------------------------------------------

Here, we would like to point out that it is simple to define the boundary conditions for which one can prove that the homogeneous version of the Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ does not admit trigonometric polynomial solutions for $\tau (\lambda )\in \Sigma _{\mathcal{T}}$.

\[impossible\_hom\] Assume that the boundary parameters satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in 
\mathbb{Z}  \label{Inhomogeneous-boundary conditions}$$where we have defined:$$Y^{(i,r)}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })\equiv \tau _{-}-\tau
_{+}+\left( -1\right) ^{i}\left[ \left( \mathsf{N}-1-r\right) \eta +(\alpha
_{-}+\alpha _{+}+\beta _{-}-\beta _{+})\right] ,$$then for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  the homogeneous Baxter equation:$$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta ),$$does not admit any (non identically zero) $Q(\lambda )$ of Laurent polynomial form in $e^{\lambda }$.

If we consider the following function:$$Q(\lambda )=\sum_{a=-s}^{r}y_{a}e^{a\lambda },\text{ \ with }r,s\in \mathbb{N}$$we can clearly always chose the coefficients $y_{a}$ such that the r.h.s. of the homogeneous Baxter equation has no poles as required. However, it is enough to consider now the asymptotics:$$\begin{aligned}
\lim_{\lambda \rightarrow +\infty }\frac{\left[ \mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )\right] }{e^{(2\mathsf{N}+4+r)\lambda }}& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\alpha
_{+}+\alpha _{-}-\beta _{+}+\beta _{-}+(\mathsf{N}-1-r)\eta )}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}} \\
\lim_{\lambda \rightarrow +\infty }e^{-(2\mathsf{N}+4+r)\lambda }\tau
(\lambda )Q(\lambda )& =\frac{y_{r}\kappa _{+}\kappa _{-}\cosh (\tau
_{+}-\tau _{-})}{2^{2(\mathsf{N}+1)}\sinh \zeta _{+}\sinh \zeta _{-}}\end{aligned}$$and use the conditions $\left( \ref{Inhomogeneous-boundary conditions}\right) $ to observe that for any $r\in \mathbb{Z}$ the asymptotic of the homogeneous Baxter equation cannot be satisfied which implies the validity of the lemma.

SOV spectrum in terms of the inhomogeneous Baxter equation
----------------------------------------------------------

We introduce now the following function of the boundary parameters:$$F_{0}=\frac{2\kappa _{+}\kappa _{-}\left( \cosh (\tau _{+}-\tau _{-})-\cosh
(\alpha _{+}+\alpha _{-}-\beta _{+}+\beta _{-}-(\mathsf{N}+1)\eta )\right) }{\sinh \zeta _{+}\sinh \zeta _{-}},$$and then the function:$$\begin{aligned}
F(\lambda ) &=&2^{\mathsf{N}}\,F_{0}\,(\cosh ^{2}2\lambda -\cosh ^{2}\eta
)a(\lambda )a(-\lambda )d(-\lambda )d(\lambda ) \\
&=&F_{0}\, (\cosh ^{2}2\lambda -\cosh ^{2}\eta )\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}(\cosh 2\lambda -\cosh 2\zeta _{b}^{(i)}).\end{aligned}$$We introduce also the set of functions $\Sigma _{\mathcal{Q}}$ such that $Q(\lambda)\in\Sigma _{\mathcal{Q}}$ if it has a form $\left( \ref{Q-form2}\right) $ and $$\tau(\lambda)=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}$$ is a trigonometric polynomial. We are now ready to prove the main theorem of this article:

\[T-eigenvalue-F-eq\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\tau _{\pm },\alpha
_{\pm },\beta _{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} ,  \label{Inhom-cond-BaxEq}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q}(\lambda )+F(\lambda ).$$

First we prove that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  then there is a trigonometric polynomial $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ satisfying the inhomogeneous functional Baxter equation: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$To prove it we will show that there is the unique set of values $Q(\zeta _{b}^{(0)})$ such that $Q(\lambda)$ of the form (\[Q-form1\]) satisfies this equation. It is straightforward to verify that if $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ and $Q(\lambda )$ has the form $\left( \ref{Q-form2}\right) $ then the left and right hand sides of the above equation are both even trigonometric polynomials of $\lambda 
$ and both can be written (using the asymptotic behavior) in the form:$$\frac{2\kappa _{+}\kappa _{-}\cosh (\tau _{+}-\tau _{-})\prod_{b=1}^{2\mathsf{N}+2}(\cosh 2\lambda -\cosh 2y_{b}^{\left( lhs/rhs\right) })}{\sinh
\zeta _{+}\sinh \zeta _{-}}.$$Then to prove that we can introduce a $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ which satisfies the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ with $\tau (\lambda )\in \Sigma _{\mathcal{T}} $, we have only to prove that $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied in $4\mathsf{N}+4$ different values of $\lambda $. As the [*r.h.s*]{} and [*l.h.s*]{} of $\left( \ref{Inhom-Baxter-Eq}\right) $ are even functions we need to check this identity only for $2N+2$ non-zero points $\mu_j$ such that $\mu_j\neq \pm \mu_k$. It is a simple exercise verify that the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ is satisfied automatically for any $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ in the following two points, $ \eta /2$ and $ \eta
/2+i\pi /2$:$$\tau (\eta /2)Q(\eta /2)=\mathbf{A}(\eta /2)Q(\eta /2-\eta )=\mathbf{A}(\eta /2)Q(\eta /2),  \label{System-A}$$and:$$\tau (\eta /2+i\pi /2)Q(\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)Q(i\pi /2-\eta /2) \\ 
=\mathbf{A}(\eta /2+i\pi /2)Q(\eta /2+i\pi /2) .  \label{System-B}$$Indeed, these equations reduce to:$$\tau (\eta /2)=\mathbf{A}(\eta /2),\text{ \ \ \ }\tau (\eta /2+i\pi /2)=\mathbf{A}(\eta /2+i\pi /2)$$and so they are satisfied by definition for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}$. Then we check the explicit form of the equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $\zeta _{b}^{(1)}$:$$\tau (\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)})=\mathbf{A}(-\zeta _{b}^{(0)})Q(\zeta _{b}^{(0)}+\eta )=\mathbf{A}(-\zeta
_{b}^{(0)})Q(\zeta _{b}^{(1)}),$$and:$$\tau (\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)})=\mathbf{A}(\zeta _{b}^{(1)})Q(\zeta _{b}^{(1)}-\eta )=\mathbf{A}(\zeta
_{b}^{(1)})Q(\zeta _{b}^{(0)}).$$They are equivalent to the following system of equations:$$\begin{aligned}
\frac{\mathbf{A}(\zeta _{b}^{(1)})}{\tau (\zeta _{b}^{(1)})}& =\frac{\tau
(\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall
b\in \{1,...,\mathsf{N}\}  \label{System1} \\
\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}& =\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1  \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})+2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta
_{a}^{(0)}\right)  \label{System2}\end{aligned}$$Now using the following quantum determinant identity $$\frac{\det_{q}K_{+}(\lambda-\eta/2)\det_{q}
\mathcal{U}_{-}(\lambda -\eta /2)}{\sinh (2\lambda +\eta )\sinh (2\lambda -\eta )}=\mathbf{A}(\lambda )\mathbf{A}(-\lambda +\eta ).\label{Tot-q-det-tt}$$ it is easy to see that the system of equations $\left( \ref{System1}\right) $ is certainly satisfied as $\tau (\lambda )\in \Sigma _{\mathcal{T}}$, once we recall the SOV characterization (\[Interpolation-Form-T\]) of $\Sigma _{\mathcal{T}}$. Indeed there is a set $\{x_1,\dots,x_n\}$ satisfying the equations (\[Quadratic System\]) and $\tau(\zeta _{b}^{(0)})=x_b$.

So we are left with $\left( \ref{System2}\right) $ a linear system of $\mathsf{N}$ inhomogeneous equations with $\mathsf{N}$ unknowns $Q(\zeta _{a}^{(0)})$. Here, we prove that the matrix of this linear system$$c_{a b}\equiv \prod_{\substack{ c=1  \\ c\neq a}}^{\mathsf{N}}\frac{\cosh
2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh 2\zeta _{a}^{(0)}-\cosh
2\zeta _{c}^{(0)}}-\delta _{a b}\frac{\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta _{b}^{(0)})}\text{ \ \ \ \ \ }\forall a,b\in \{1,...,\mathsf{N}\}$$ has nonzero determinant for the given $\tau (\lambda
)\in \Sigma _{\mathcal{T}}$. Indeed, let us suppose that for some $\tau (\lambda )\in \Sigma _{\mathcal{T}}$:$$\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] =0.  \label{det-coeff}$$Then there is at least one nontrivial solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}\neq \{0,...,0\}$ to the homogeneous system of equations:$$\frac{Q(\zeta _{b}^{(0)})\tau (\zeta _{b}^{(0)})}{\mathbf{A}(-\zeta
_{b}^{(0)})}=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ c=1  \\ c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta _{b}^{(1)}-\cosh 2\zeta _{c}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{c}^{(0)}}Q(\zeta _{a}^{(0)})
\label{System2-homo}$$and hence we can define:$$Q_{\mathsf{M}}(\lambda )=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{ b=1  \\ b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta _{b}^{(0)}}{\cosh
2\zeta _{a}^{(0)}-\cosh 2\zeta _{b}^{(0)}}Q(\zeta _{a}^{(0)})=\lambda _{\mathsf{M}+1}^{(\mathsf{M})}\prod_{b=1}^{\mathsf{M}}\left( \cosh 2\lambda
-\cosh 2\lambda _{b}^{(\mathsf{M})}\right) .$$It is an even trigonometric polynomial of degree $2\mathsf{M}$ such that $0\leq 
\mathsf{M}\leq \mathsf{N}-1$ fixed by the solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$. Now using the $Q_{\mathsf{M}}(\lambda )$ and $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we can define two functions:$$W_{1}(\lambda )=Q_{\mathsf{M}}(\lambda )\tau (\lambda )\text{ \, and \, }W_{2}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )$$which are both even trigonometric polynomials of degree $2\mathsf{M}+2\mathsf{N}+4$. Then it is straightforward to observe that the systems of equations $\left( \ref{System1}\right) $ and $\left( \ref{System2-homo}\right) $ plus the conditions $\left( \ref{System-A}\right) $ and $\left( \ref{System-B}\right) 
$, which are also satisfied with the function $Q_{\mathsf{M}}(\lambda )$, imply that $W_{1}(\lambda )$ and $W_{2}(\lambda )$ coincide in $4\mathsf{N}+4 $ different values of $\lambda $ ($\pm \eta /2$, $\pm (\eta /2+i\pi /2)$, $\pm \zeta _{b}^{(0)}$ and $\pm \zeta _{b}^{(1)}$). It means that $W_{1}(\lambda )\equiv W_{2}(\lambda )$, as these are two polynomials of maximal degree $4\mathsf{N}+2$. So, we have shown that from the assumption $\exists \tau (\lambda )\in \Sigma _{\mathcal{T}}$ such that $\left( \ref{det-coeff}\right) $ holds it follows that $\tau (\lambda )$ and $Q_{\mathsf{M}}(\lambda )$ have to satisfy the following homogeneous Baxter equations: $$\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ).
\label{Baxter-eq-homo}$$Now we can apply the Lemma \[impossible\_hom\] which implies that $Q_{\mathsf{M}}(\lambda )=0$ for any $\lambda$, which contradicts the hypothesis of the existence of a nontrivial solution to the homogeneous system [(\[System2-homo\])]{}. Hence, we have proven that $\mathrm{det}_{\mathsf{N}}\left[ c_{a b}\right] \neq 0.$ Therefore there is a unique solution $\{Q(\zeta
_{1}^{(0)}),...,Q(\zeta _{\mathsf{N}}^{(0)})\}$ of the inhomogeneous system $\left( \ref{System2}\right) $ which defines one and only one $Q(\lambda )$ of the form $\left( \ref{Q-form1}\right) $ satisfying the functional inhomogeneous Baxter’s equation $\left( \ref{Inhom-Baxter-Eq}\right) $.

We prove now that if $Q(\lambda )\in \Sigma _{\mathcal{Q}}$ then $\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )\in
\Sigma _{\mathcal{T}}$. By definition of the functions $Z_{Q}(\lambda ),$ $F(\lambda )$ and $Q(\lambda )$ the function $\tau (\lambda )$ has the desired form:$$\tau (\lambda )=f(\lambda )+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )\tau
(\zeta _{a}^{(0)}).$$To prove now that $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ we have to write the inhomogeneous Baxter equation $\left( \ref{Inhom-Baxter-Eq}\right) $ in the $2\mathsf{N}$ points $ \zeta _{b}^{(0)}$ and $ \zeta
_{b}^{(1)}$. Indeed, we have already proved that this reproduce the systems $\left( \ref{System1}\right) $ and $\left( \ref{System2}\right) $ and it is simple to observe that the system of equations $\left( \ref{System1}\right) $ just coincides with the inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$once we define $x_{a}=\tau (\zeta _{a}^{(0)})$ for any $a\in \{1,...,\mathsf{N}\}$ and we write $\tau (\zeta _{n}^{(1)})$ in terms of the $x_{a}$. Thus we show that $$\tau (\lambda )=\left( Z_{Q}(\lambda )+F(\lambda )\right)
/Q(\lambda )\in \Sigma _{\mathcal{T}},$$ completing the proof of the theorem.

Completeness of the Bethe ansatz equations
------------------------------------------

In the previous section we have shown that to solve the transfer matrix spectral problem associated to the most general representations of the trigonometric 6-vertex reflection algebra we have just to classify the set of functions $Q(\lambda )$ of the form $\left( \ref{Q-form2}\right) $ for which $\left( Z_{Q}(\lambda )+F(\lambda )\right) /Q(\lambda )$ is a trigonometric polynomial; i.e. the set of functions $\Sigma _{\mathcal{Q}}$ completely fixes the set $\Sigma _{\mathcal{T}}$. We can show now that the previous characterization of the transfer matrix spectrum allows to prove that $\Sigma _{InBAE}\subset \mathbb{C}^{\mathsf{N}}$ the set of all the solutions of inhomogeneous Bethe equations $$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} ,  \label{I-BAE}$$ defines the complete set of transfer matrix eigenvalues. In particular, the following corollary follows:

\[Theo-InBAE\] Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic $\left( \ref{xi-conditions}\right) $ and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy $\left( \ref{Inhom-cond-BaxEq}\right) $ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh
2\lambda -\cosh 2\lambda _{a}\right) .$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{InBAE}$ of all the solutions to the inhomogeneous system of Bethe equations $\left( \ref{I-BAE}\right) $ contains $2^{\mathsf{N}}$ elements.

Homogeneous Baxter equation
===========================

Boundary conditions annihilating the inhomogeneity of the Baxter equation
-------------------------------------------------------------------------

The description presented in the previous sections can be applied to completely general integrable boundary terms including as a particular case the boundary conditions for which the inhomogeneous term in the functional Baxter equation vanishes. As these are still quite general boundary conditions it is interesting to point out how the previous general results explicitly look like in these cases.

\[homogeneousBE\_N\] Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in 
\mathbb{C}^{6}\backslash N_{SOV}$ satisfying the condition:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} \text{\ }:Y^{(i,2\mathsf{N})}(\tau _{\pm },\alpha _{\pm },\beta
_{\pm })=0  \label{ond-homo-boundary}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $ \exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda
)Q(\lambda +\eta ).$$Or equivalently, $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )}\text{ \ \ with \ \ }Q(\lambda
)=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh
2\lambda _{a}\right) .$$where:$$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} .  \label{BAE}$$Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.

This theorem is just a rewriting of the results presented in the Theorem \[T-eigenvalue-F-eq\] and Corollary \[Theo-InBAE\] for the case of vanishing inhomogeneous term. Indeed if the conditions $\left( \ref{BAE}\right) $ are satisfied then automatically the conditions of the main theorem $\left( \ref{Inhom-cond-BaxEq}\right) $ are satisfied too that implies that the map from the $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  to the $\{\lambda
_{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ is indeed an isomorphism.

More general boundary conditions compatibles with homogeneous Baxter equations
------------------------------------------------------------------------------

We address here the problem of describing the boundary conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} :Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm },\beta _{\pm })=0,
\label{Cond-homo-M}$$for which the conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ are not satisfied and then the Theorem \[T-eigenvalue-F-eq\] cannot be directly applied. In these $2\mathsf{N}$ hyperplanes in the space of the boundary parameters we have just to modify this theorem to take into account that the Baxter equation associated to the choice of coefficient $\mathbf{A}(\lambda
) $ is indeed compatible with the homogeneous Baxter equation for a special choice of the polynomial $Q(\lambda )$. First we define the following functions$$Q_{\mathsf{M}}(\lambda )=2^{\mathsf{M}}\prod_{b=1}^{\mathsf{M}}\left( \cosh
2\lambda -\cosh 2\lambda _{b}^{(\mathsf{M})}\right) .  \label{Q-form-M}$$We introduce also the set of polynomials $\Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ if $Q_{\mathsf{M}}(\lambda )$ has a form $\left( \ref{Q-form-M}\right)$ and $$\tau (\lambda )=\frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}$$is a trigonometric polynomial. Then we can define the corresponding set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$$$\Sigma _{\mathcal{T}}^{\mathsf{M}}=\left\{ \tau (\lambda ):\tau (\lambda
)\equiv \frac{\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta )}{Q_{\mathsf{M}}(\lambda )}\text{
\, if  }Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}\right\}
.$$It is simple to prove the validity of the following:

\[mixed-condition\] Let the boundary conditions $\left( \ref{Cond-homo-M}\right) $ be satisfied, then $\Sigma _{\mathcal{T}}^{\mathsf{M}}\subset\Sigma _{\mathcal{T}}$ and moreover for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q_{\mathsf{M}}(\lambda )\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ such that:$$\label{homogen-Bax-eq-M}
\tau (\lambda )Q_{\mathsf{M}}(\lambda )=\mathbf{A}(\lambda )Q_{\mathsf{M}}(\lambda -\eta )+\mathbf{A}(-\lambda )Q_{\mathsf{M}}(\lambda +\eta ),$$and for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}\backslash \Sigma _{\mathcal{T}}^{\mathsf{M}}$ there exists one and only one $Q(\lambda )\in
\Sigma _{\mathcal{Q}}$ such that:$$\label{inhomogen-Bax-eq-M}
\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda ).$$

The proof follows the one given for the main Theorem \[T-eigenvalue-F-eq\] we have just to observe that thanks to the boundary conditions $\left( \ref{Cond-homo-M}\right) $ the set $\Sigma _{\mathcal{T}}^{\mathsf{M}}$ is formed by transfer matrix eigenvalues as the Baxter equation implies that for any $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ the systems of equations $\left( \ref{System-A}\right) ,$ $\left( \ref{System-B}\right) $ and $\left( \ref{System1}\right) $ are satisfied and moreover that the asymptotics of the $\tau (\lambda )\in \Sigma _{\mathcal{T}}^{\mathsf{M}}$ is exactly that of the transfer matrix eigenvalues.

Finally, it is interesting to remark that under the boundary conditions $\left( \ref{Inhom-cond-BaxEq}\right) $ the complete characterization of the spectrum of the transfer matrix is given in terms of the even polynomials $Q(\lambda )$ all of fixed degree $2\mathsf{N}$ and form $\left( \ref{Q-form2}\right) $ which are solutions of the inhomogeneous/homogeneous Baxter equation. However, in the cases when the boundary parameters satisfy the constraints $\left( \ref{Cond-homo-M}\right) $ for a given $\mathsf{M}\in \left\{ 0,...,\mathsf{N}-1\right\} $ a part of the transfer matrix spectrum can be defined by polynomials of smaller degree; i.e. the $Q_{\mathsf{M}}(\lambda
)\in \Sigma _{\mathcal{Q}}^{\mathsf{M}}$ for the fixed $\mathsf{M}\in
\left\{ 0,...,\mathsf{N}-1\right\} $.

Discrete symmetries and equivalent Baxter equations {#sect-descretesym}
===================================================

It is important to point out that we have some large amount of freedom in the choice of the functional reformulation of the SOV characterization of the transfer matrix spectrum. We have reduced it looking for trigonometric polynomial solutions $Q(\lambda )$ of the second order difference equations with coefficients $\mathbf{A}(\lambda )$ which are rational trigonometric functions. It makes the finite difference terms $\mathbf{A}(\lambda
)Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )$ in the functional equation a trigonometric polynomial. Indeed, this assumption reduces the possibility to use the following gauge transformations of the coefficients allowed instead by the SOV characterization:$$\mathbf{A}_{\alpha }(\lambda )=\alpha (\lambda )\mathbf{A}(\lambda ),\text{
\ }\mathbf{D}_{\alpha }(\lambda )=\frac{\mathbf{A}(-\lambda )}{\alpha
(\lambda +\eta )}.$$In the following we discuss simple transformations that do not modify the functional form of the coefficients allowing equivalent reformulations of the SOV spectrum by Baxter equations.

Discrete symmetries of the transfer matrix spectrum
---------------------------------------------------

It is not difficult to see that the spectrum (eigenvalues) of the transfer matrix presents the following invariance:

\[Lem-invariance\]We denote explicitly the dependence from the boundary parameters in the set of boundary parameters $\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha
_{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})}$ of the eigenvalue functions of the transfer matrix $\mathcal{T}(\lambda )$, then this set is invariant under the following $Z_{2}^{\otimes 3}$ transformations of the boundary parameters:$$\begin{aligned}
&\Sigma _{\mathcal{T}}^{(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha
_{-},\beta _{-})}\equiv \Sigma _{\mathcal{T}}^{(\epsilon _{\tau }\tau
_{+},\epsilon _{\alpha }\alpha _{+},\epsilon _{\beta }\beta _{+},\epsilon
_{\tau }\tau _{-},\epsilon _{\alpha }\alpha _{-},\epsilon _{\beta }\beta
_{-})}\ \\ &
\forall (\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}.\nonumber\end{aligned}$$

To prove this statement it is enough to look at the SOV characterization which defines completely the transfer matrix spectrum, i.e. the set $\Sigma _{\mathcal{T}}$, and to prove that it is invariant under the above considered $Z_{2}^{\otimes 3}$ transformations of the boundary parameters. We have first to remark that the central values $\left( \ref{Central-asymp}\right) $-$\left( \ref{Central-2}\right) $ of the transfer matrix $\mathcal{T}(\lambda )$ are invariant under these discrete transformations and then the function $f(\lambda )$, defined in $\left( \ref{f-function}\right) $, is invariant too and the same is true for the form $\left( \ref{Interpolation-Form-T}\right) $ of the interpolation polynomial describing the elements of $\Sigma _{\mathcal{T}}$. Then the invariance of the SOV characterization $\left( \ref{Quadratic System}\right) $ follows from the invariance of the quantum determinant$$\begin{aligned}
\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )
&=&\sinh (2\eta -2\lambda )\sinh (2\lambda +2\eta )g_{+}(\lambda +\eta
/2)g_{+}(-\lambda +\eta /2)g_{-}(\lambda +\eta /2)  \notag \\
&&\times g_{-}(-\lambda +\eta /2)a(\lambda +\eta /2)d(\lambda -\eta
/2)a(-\lambda +\eta /2)d(-\lambda -\eta /2)\end{aligned}$$ under these discrete transformations.

It is important to underline that the above $Z_{2}^{\otimes 3}$ transformations of the boundary parameters do indeed change the transfer matrix $\mathcal{T}(\lambda )$ and the Hamiltonian and so this invariance is equivalent to the statement that these different transfer matrices are all isospectral. In particular, it is simple to find the similarity matrices implementing the following $Z_{2}$ transformations of the boundary parameters:$$\begin{aligned}
\mathcal{T}(\lambda |-\tau _{+},-\zeta _{+},\kappa _{+},-\tau _{-},-\zeta
_{-},\kappa _{-}) &=&\Gamma _{y}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{y},\text{ \ \ \ }\Gamma _{y}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{y}, \\
\mathcal{T}(\lambda |\tau _{+},\zeta _{+},-\kappa _{+},\tau _{-},\zeta
_{-},-\kappa _{-}) &=&\Gamma _{z}\mathcal{T}(\lambda |\tau _{+},\zeta
_{+},\kappa _{+},\tau _{-},\zeta _{-},\kappa _{-})\Gamma _{z},\text{ \ \ \ }\Gamma _{z}\equiv \otimes _{n=1}^{\mathsf{N}}\sigma _{n}^{z}.\end{aligned}$$

Equivalent Baxter equations and the SOV spectrum
------------------------------------------------

The invariance of the spectrum $\Sigma _{\mathcal{T}}$ under these $Z_{2}^{\otimes 3}$ transformations of the boundary parameters can be used to define equivalent Baxter equation reformulation of $\Sigma _{\mathcal{T}}$. More precisely, let us introduce the following functions $\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )$ and $F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )$ obtained respectively by implementing the $Z_{2}^{\otimes 3}$ transformations:$$(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\rightarrow (\epsilon _{\tau }\tau _{+},\epsilon _{\alpha }\alpha
_{+},\epsilon _{\beta }\beta _{+},\epsilon _{\tau }\tau _{-},\epsilon
_{\alpha }\alpha _{-},\epsilon _{\beta }\beta _{-}),$$then the following characterizations hold for any fixed $(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times
\{-1,1\}$:

\[T-eigenvalue-F-eq-gen\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{} and let the boundary parameters $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash N_{SOV}$ satisfy the following conditions:$$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }Y^{(i,2r)}(\epsilon _{\tau
}\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm },\epsilon _{\beta }\beta
_{\pm })\neq 0\text{ \ }\forall i\in \left\{ 0,1\right\} ,r\in \left\{ 0,...,\mathsf{N}-1\right\} ,  \label{Inhom-cond-BaxEq-gen}$$then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ),$$where:$$Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta ).$$

The proof follows step by step the one given for the main Theorem \[T-eigenvalue-F-eq\].

General validity of the inhomogeneous Baxter equations
------------------------------------------------------

The previous reformulations of the spectrum in terms of different inhomogeneous Baxter equations and the observation that the conditions under which the Theorem does not apply are related to the choice of the $(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}\times
\{-1,1\}\times \{-1,1\}$ allow us to prove that unless the boundary parameters are lying on a finite lattice of step $\eta $ we can always use an inhomogeneous Baxter equations to completely characterize the spectrum of the transfer matrix. More precisely, let us introduce the following hyperplanes in the space of the boundary parameters:$$M\equiv \left\{ 
\begin{array}{l}
(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in 
\mathbb{C}^{6}:\exists (r_{+,+},r_{-,+},r_{-,-})\in \{0,...,\mathsf{N}-1\}
\\ 
\text{ such that: \ \ \ }\left\{ 
\begin{array}{l}
r_{+,+}+r_{-,-}-r_{-,+}\in \{0,...,\mathsf{N}-1\} \\ 
\alpha _{+}+\alpha _{-}=(r_{-,+}-r_{+,+})\eta \\ 
\beta _{-}-\beta _{+}=(r_{-,-}-r_{-,+})\eta \\ 
\tau _{-}-\tau _{+}=(\mathsf{N}-1+r_{-,-}-3r_{+,+})\eta\end{array}\right.\end{array}\right\}  \label{Def-M}$$then the following theorem holds:

Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the conditions [(\[xi-conditions\])]{} and let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}\backslash \left( M\cup N_{SOV}\right) $ then we can always find a $(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta })\in
\{-1,1\}\times \{-1,1\}\times \{-1,1\}$ such that $\tau (\lambda )\in\Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=Z_{Q,(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda )+F_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon _{\beta
})}(\lambda ).$$

The Theorem \[T-eigenvalue-F-eq-gen\] does not apply if $\exists i\in
\left\{ 0,1\right\} $ and $\exists r\in \left\{ 0,...,\mathsf{N}-1\right\} $ such that the following system of conditions on the boundary parameters are satisfied:$$Y^{(i,2r)}(\epsilon _{\tau }\tau _{\pm },\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\text{ \ }\forall (\epsilon _{\tau
},\epsilon _{\alpha },\epsilon _{\beta })\in \{-1,1\}^{\otimes 3}
\label{Cond-general-homo}$$ then by simple computations it is possible to observe that the set $M$ defined in $\left( \ref{Def-M}\right) $ indeed coincides with the following set:$$\left\{ (\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta
_{-})\in \mathbb{C}^{6}:\text{ }\exists i\in \left\{ 0,1\right\} ,r\in
\left\{ 0,...,\mathsf{N}-1\right\} \text{ such that }\left( \ref{Cond-general-homo}\right) \text{ is satisfied}\right\} ,$$from which the theorem clearly follows.

Homogeneous Baxter equation
---------------------------

The discrete symmetries of the transfer matrix allow also to define the general conditions on the boundary parameters for which the spectrum can be characterized by a homogeneous Baxter equation. In particular the following corollary holds:

Let $(\tau _{+},\alpha _{+},\beta _{+},\tau _{-},\alpha _{-},\beta _{-})\in 
\mathbb{C}^{6}\backslash N_{SOV}$ satisfy the condition:$$\begin{aligned}
&\kappa _{+}\neq 0,\kappa _{-}\neq 0,\nonumber\\
&\exists i\in \left\{
0,1\right\} ,\text{\ }\exists (\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })\in \{-1,1\}\times \{-1,1\}\times \{-1,1\}:Y^{(i,2\mathsf{N})}(\epsilon _{\tau }\tau _{\pm },,\epsilon _{\alpha }\alpha _{\pm
},\epsilon _{\beta }\beta _{\pm })=0\end{aligned}$$and let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}
$ $^{\mathsf{N}}$ be generic [(\[xi-conditions\])]{}, then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$ if and only if $\exists
!Q(\lambda )\in \Sigma _{\mathcal{Q}}$ such that $$\tau (\lambda
)Q(\lambda )=\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha },\epsilon
_{\beta })}(\lambda )(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta
).$$Or equivalently we can define the set of all the solutions of the Bethe equations $$\Sigma _{BAE}=\left\{ \{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \mathbb{C}^{\mathsf{N}}:\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}+\eta
)=0,\text{ \ }\forall a\in \{1,...,\mathsf{N}\}\right\} .$$Then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  if and only if $\exists !\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{BAE}$ such that:$$\tau (\lambda )=\frac{\mathbf{A}_{(\epsilon _{\tau },\epsilon _{\alpha
},\epsilon _{\beta })}(\lambda )Q(\lambda -\eta )+\mathbf{A}_{(\epsilon
_{\tau },\epsilon _{\alpha },\epsilon _{\beta })}(-\lambda )Q(\lambda +\eta )}{Q(\lambda )},$$with $$Q(\lambda )=2^{\mathsf{N}}\prod_{a=1}^{\mathsf{N}}\left( \cosh 2\lambda -\cosh 2\lambda _{a}\right).$$ Moreover, under the condition of normality defined in Proposition [normality]{}, the set $\Sigma _{BAE}$ of the solutions to the homogeneous system of Bethe ansatz type equations $\left( \ref{BAE}\right) $ contains $2^{\mathsf{N}}$ elements.

XXX chain by SOV and Baxter equation
====================================

The construction of the SOV characterization can be naturally applied in the case of the rational 6-vertex $R$-matrix, which in the homogeneous limit reproduces the XXX open quantum spin-1/2 chain with general integrable boundary conditions[^4]. Let us define:$$R_{12}(\lambda )=\left( 
\begin{array}{cccc}
\lambda +\eta & 0 & 0 & 0 \\ 
0 & \lambda & \eta & 0 \\ 
0 & \eta & \lambda & 0 \\ 
0 & 0 & 0 & \lambda +\eta\end{array}\right) \in \text{End}(\mathcal{H}_{1}\otimes \mathcal{H}_{2}).$$Due to the $SU(2)$ invariance of the bulk monodromy matrix the boundary matrices defining the most general integrable boundary conditions can be always recasted in the following form:$$K_{-}(\lambda ;p)=\left( 
\begin{array}{cc}
\lambda -\eta /2+p & 0 \\ 
0 & p-\lambda +\eta /2\end{array}\right) ,\text{ \ \ \ }K_{+}(\lambda ;q,\xi )=\left( 
\begin{array}{cc}
\lambda +\eta /2+q & \xi (\lambda +\eta /2) \\ 
\xi (\lambda +\eta /2) & q-(\lambda +\eta /2)\end{array}\right) ,$$leaving only three arbitrary complex parameters here denoted with $\xi ,$ $p$ and $q$. Then the one parameter family of commuting transfer matrices:$$\mathcal{T}(\lambda )=\text{tr}_{0}\{K_{+}(\lambda )\,M(\lambda
)\,K_{-}(\lambda )\hat{M}(\lambda )\}\in \text{\thinspace End}(\mathcal{H}),$$in the homogeneous limit leads to the following Hamiltonian:$$H=\sum_{n=1}^{\mathsf{N}}\left( \sigma _{n}^{x}\sigma _{n+1}^{x}+\sigma
_{n}^{y}\sigma _{n+1}^{y}+\sigma _{n}^{z}\sigma _{n+1}^{z}\right) +\frac{\sigma _{\mathsf{N}}^{z}}{p}+\frac{\sigma _{1}^{z}+\xi \sigma _{1}^{x}}{q}.$$It is simple to show that the following identities hold:$$\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda
)=4(\lambda ^{2}-\eta ^{2})(\lambda ^{2}-p^{2})((1+\xi ^{2})\lambda
^{2}-q^{2})\prod_{b=1}^{\mathsf{N}}(\lambda ^{2}-(\xi _{n}+\eta
)^{2})(\lambda ^{2}-(\xi _{n}-\eta )^{2}).$$We define:$$\mathbf{A}(\lambda )=(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }(\lambda -\eta /2+p)(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q)\prod_{b=1}^{\mathsf{N}}(\lambda -\zeta _{b}^{(0)})(\lambda +\zeta _{b}^{(1)}),$$then it is easy to derive the following quantum determinant identity:$$\frac{\mathrm{det}_{q}K_{+}(\lambda )\mathrm{det}_{q}\,\mathcal{U}_{-}(\lambda )}{(4\lambda ^{2}-\eta ^{2})}=\mathbf{A}(\lambda +\eta /2)\mathbf{A}(-\lambda
+\eta /2).$$ From the form of the boundary matrices it is clear that for the rational 6-vertex case one can directly derive the SOV representations using the method developed in [@Nic12b] without any need to introduce Baxter’s gauge transformations. Some results in this case also appeared in [@FraSW08; @FraGSW11] based on a functional version of the separation of variables of Sklyanin, a method which allows to define the eigenvalues and wave-functions but which does not allow to construct in the original Hilbert space of the quantum chain the transfer matrix eigenstates.

The separation of variable description in this rational 6-vertex case reads:

\[C:T-eigenstates- copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic: $$\xi _{a}\neq \pm\xi _{b}+r\eta \text{ \ }\forall a\neq b\in \{1,...,\mathsf{N}\}\,\,\text{and\thinspace \thinspace }r\in \{-1,0,1\},  \label{xi-conditions-xxx}$$ then $\mathcal{T}(\lambda )$ has simple spectrum and $\Sigma _{\mathcal{T}}$ is characterized by:$$\Sigma _{\mathcal{T}}=\left\{ \tau (\lambda ):\tau (\lambda )=f(\lambda
)+\sum_{a=1}^{\mathsf{N}}g_{a}(\lambda )x_{a},\text{ \ \ }\forall
\{x_{1},...,x_{\mathsf{N}}\}\in \Sigma _{T}\right\} ,$$where:$$g_{a}(\lambda )=\frac{4\lambda ^{2}-\eta ^{2}}{4{\zeta
_{a}^{(0)}} ^{2}-\eta ^{2}}\,\prod_{\substack{ b=1  \\ b\neq a}}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta _{b}^{(0)}} ^{2}}{{
\zeta _{a}^{(0)}} ^{2}-{ \zeta _{b}^{(0)}} ^{2}}\quad \text{
\ for }a\in \{1,...,\mathsf{N}\},$$and$$f(\lambda )=\prod_{b=1}^{\mathsf{N}}\frac{\lambda ^{2}-{ \zeta
_{b}^{(0)}}^{2}}{{ \zeta _{a}^{(0)}} ^{2}-{\zeta
_{b}^{(0)}} ^{2}}\mathbf{A}(\eta /2)+2\left( 4\lambda ^{2}-\eta
^{2}\right) \,\prod_{b=1}^{\mathsf{N}}\lambda ^{2}-{ \zeta
_{b}^{(0)}} ^{2},$$$\Sigma _{T}$ is the set of solutions to the following inhomogeneous system of $\mathsf{N}$ quadratic equations:$$x_{n}\sum_{a=1}^{\mathsf{N}}g_{a}(\zeta _{n}^{(1)})x_{a}+x_{n}f(\zeta
_{n}^{(1)})=q_{n},\text{ \ \ \ }q_{n}=\frac{\mathrm{det}_{q}K_{+}(\xi _{n})\mathrm{det}_{q}\,\mathcal{U}_{-}(\xi _{n})}{\eta -4\xi _{n}^{2}},\text{ \ \ }\forall n\in \{1,...,\mathsf{N}\},$$in $\mathsf{N}$ unknowns $\{x_{1},...,x_{\mathsf{N}}\}$.

We are now ready to present the following equivalent characterization of the transfer matrix spectrum:

\[T-eigenvalue-F-eq copy(1)\]Let the inhomogeneities $\{\xi _{1},...,\xi
_{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ be generic [(\[xi-conditions-xxx\])]{}, then for $\xi \neq 0$ the set of transfer matrix eigenvalue functions $\Sigma _{\mathcal{T}}$ is characterized by:$$\tau (\lambda )\in \Sigma _{\mathcal{T}}\text{ \ if and only if }\exists
!Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) \text{ such that }\tau (\lambda )Q(\lambda )=Z_{Q}(\lambda
)+F(\lambda ),$$with$$F(\lambda )=2(1-\sqrt{(1+\xi ^{2})})\left( 4\lambda ^{2}-\eta ^{2}\right)
\,\prod_{b=1}^{\mathsf{N}}\prod_{i=0}^{1}\left( \lambda ^{2}- {�\zeta
_{b}^{(i)}} ^{2}\right) .$$

The proof presented in Theorem \[T-eigenvalue-F-eq\] applies with small modifications also to present rational case.

The previous characterization of the transfer matrix spectrum allows to prove that the set $\Sigma _{InBAE}\subset\mathbb{C}^\mathsf{N}$ of all the solutions of the Bethe equations$$\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in\Sigma _{InBAE}$$ if $$\mathbf{A}(\lambda _{a})Q_{\mathbf{\lambda }}(\lambda _{a}-\eta )+\mathbf{A}(-\lambda _{a})Q_{\mathbf{\lambda }}(\lambda
_{a}+\eta )=-F(\lambda _{a}),\text{ \ }\forall a\in \{1,...,\mathsf{N}\} ,  \label{I-BAE-XXX}$$ define the complete set of transfer matrix eigenvalues. In particular, the following corollary can be proved:

\[Theo-InBAE-XXX\]Let the inhomogeneities $\{\xi _{1},...,\xi _{\mathsf{N}}\}\in \mathbb{C}$ $^{\mathsf{N}}$ satisfy the following conditions [(\[xi-conditions\])]{}, then $\mathcal{T}(\lambda )$ has simple spectrum and for $\xi \neq 0$ then $\tau (\lambda )\in \Sigma _{\mathcal{T}}$  if and only if $\exists
!\{\lambda _{1},...,\lambda _{\mathsf{N}}\}\in \Sigma _{InBAE}$ such that:$$\tau (\lambda )=\frac{Z_{Q}(\lambda )+F(\lambda )}{Q(\lambda )}\text{ \ \
with \ \ }Q(\lambda )=\prod_{b=1}^{\mathsf{N}}\left( \lambda ^{2}-\lambda
_{b}^{2}\right) .$$

Homogeneous chains and existing numerical analysis
==================================================

It is important to stress that the spectrum construction together with the corresponding statements of completeness presented in this paper strictly work for the most general spin 1/2 representations of the 6-vertex reflection algebra only for generic inhomogeneous chains. However, it is worth mentioning that the transfer matrix as well as the coefficients and the inhomogeneous term in our functional equation characterization of the SOV spectrum are analytic functions of the inhomogeneities $\{\xi _{j}\}$ so we can take without any problem the homogeneous limit ($\xi _{a}\rightarrow
0$ $\forall a\in \{1,...,\mathsf{N}\}$) in the functional equations. The main problem to be addressed then is the completeness of the description by this functional equations. Some first understanding of this central question can be derived looking at the numerical analysis [@Nep-R-2003; @Nep-R-2003add] of the completeness of Bethe Ansatz equations when the boundary constraints are satisfied and for the open XXX chain with general boundary terms [@Nep-2013].

Comparison with numerical results for the XXZ chain
---------------------------------------------------

The numerical checks of the completeness of Bethe Ansatz equations for the open XXZ quantum spin 1/2 chains were first done in [@Nep-R-2003] for the chains with non-diagonal boundaries satisfying boundary constraints: $$\kappa _{+}\neq 0,\kappa _{-}\neq 0,\text{ \ }\exists i\in \left\{
0,1\right\} ,\mathsf{M}\in \mathbb{N}\text{\ }:Y^{(i,2\mathsf{M})}(\tau _{\pm },\alpha _{\pm
},\beta _{\pm })=0.$$Indeed, under these conditions some generalizations of algebraic Bethe Ansatz can be used and so the corresponding Bethe equations can be defined.

In particular, the Nepomechie-Ravanini’s numerical results reported in [Nep-R-2003,Nep-R-2003add]{} suggest that the Bethe ansatz equations $\left( \ref{BAE}\right) $ in the homogeneous limit for the roots of the $Q$ function:$$Q(\lambda )=2^{\mathsf{M}}\prod_{a=1}^{\mathsf{M}}\left( \cosh 2\lambda -\cosh 2\lambda
_{a}\right) ,$$with the degree $\mathsf{M}$ obtained from the boundary constraint

-   for $\mathsf{M}=\mathsf{N}$ they define the complete transfer matrix spectrum.

-   for $\mathsf{M}<\mathsf{N}$ the complete spectrum of the transfer matrix contains two parts described by different Baxter equations. The first one has trigonometric polynomial solutions of degree $2\mathsf{M}$ the second one has a trigonometric polynomial solutions of degree $2\mathsf{N}-2-2\mathsf{M}$.

-   for $\mathsf{M}>\mathsf{N}$ the complete spectrum of the transfer matrix spectrum plus $\tau (\lambda )$ functions which do not belong to the spectrum of the transfer matrix.

These results seem to be compatible with our characterization for the inhomogeneous chains. Indeed, the case $\mathsf{M}=\mathsf{N}$ coincides with the case in which our Baxter functional equation becomes homogeneous. Theorem \[homogeneousBE\_N\] states that in this case for generic inhomogeneities the Bethe ansatz is complete so we can expect (from the numerical analysis) that completeness will survive in the homogeneous limit. In the case $\mathsf{M}<$, our description of the spectrum by Lemma [mixed-condition]{} separates the spectrum in two parts. A first part of the spectrum is described by trigonometric polynomial solutions of degree $2\mathsf{M}$ to the homogeneous Baxter equation [(\[homogen-Bax-eq-M\])]{} and a second part is instead described by trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{}. However, by implementing the following discrete symmetry transformations $\alpha _{\pm
}\rightarrow -\alpha _{\pm }$, $\beta _{\pm }\rightarrow -\beta _{\pm }$, $\tau _{\pm }\rightarrow -\tau _{\pm }$ and applying the same Lemma [mixed-condition]{} w.r.t. the Baxter equations with coefficients $\mathbf{A}_{(-,-,-)}(\lambda )$ we get an equivalent description of the spectrum separated in two parts. One part of the spectrum is described in terms of the solutions of the transformed homogeneous Baxter equation which should be trigonometric polynomials of degree $2\mathsf{M}^{\prime }$, with $\mathsf{M}^{\prime }=\mathsf{N}-1-\mathsf{M}$ and the second part by the inhomogeneous Baxter equation. The comparison with the numerical results then suggests that, at least in the limit of homogeneous chains, the part of the spectrum generated by the trigonometric polynomial solutions of degree 2 of the inhomogeneous Baxter equation [(\[inhomogen-Bax-eq-M\])]{} coincides with the part generated by the trigonometric polynomial solutions of degree $2\mathsf{M}^{\prime }$ of the transformed homogeneous Baxter equation.

Finally, in the case $\mathsf{M}>\mathsf{N}$ we have a complete characterization of the spectrum given by an inhomogeneous Baxter functional equation however nothing prevent to consider solutions to the homogeneous Baxter equation once we take the appropriate $Q$-function with $\mathsf{M}>\mathsf{N}$ Bethe roots. The numerical results however seem to suggest that considering the homogeneous Baxter equations is not the proper thing to do in the homogeneous limit.

The previous analysis seems to support the idea that in the limit of homogeneous chain our complete characterization still describe the complete spectrum of the homogeneous transfer matrix.

Comparison with numerical results for the XXX chain
---------------------------------------------------

In the case of the open spin 1/2 XXX chain an ansatz based on two $Q$-functions and an inhomogeneous Baxter functional equation has been first introduced in [@CaoYSW13-2], the completeness of the spectrum obtained by that ansatz has been later verified numerically for small chains [@CaoJYW2013]. Using these results Nepomechie has introduced a simpler ansatz and developed some further numerical analysis in [@Nep-2013] confirming once again that the ansatz defines the complete spectrum for small chains. Here, we would like to point out that our complete description of the transfer matrix spectrum in terms of a inhomogeneous Baxter functional equation obtained for the inhomogeneous chains has the following well defined homogeneous limit: $$\tau (\lambda )Q(\lambda )=\mathbf{A}(\lambda )Q(\lambda -\eta )+\mathbf{A}(-\lambda )Q(\lambda +\eta )+F(\lambda )$$where:$$\begin{aligned}
F(\lambda ) &=&8(1-\sqrt{(1+\xi ^{2})})\left( \lambda ^{2}-\left( \eta
/2\right) ^{2}\right) ^{2\mathsf{N}+1}, \\
\mathbf{A}(\lambda ) &=&(-1)^{\mathsf{N}}\frac{2\lambda +\eta }{2\lambda }\left(\vphantom{\sqrt{(1+\xi ^{2})}}\lambda -\eta /2+p\right)\left(\sqrt{(1+\xi ^{2})}(\lambda -\eta /2)+q\right)\left(\lambda ^{2}-\left( \eta /2\right) ^{2}\right)^{\mathsf{N}}.\end{aligned}$$Taking into account the shift in our definition of the monodromy matrix which insures that the transfer matrix is an even function of the spectral parameter, the limit of our inhomogeneous Baxter functional equation coincides with the ansatz proposed by Nepomechie in [@Nep-2013]. Then the numerical evidences of completeness derived by Nepomechie in [@Nep-2013] suggest that the exact and complete characterization that we get for the inhomogeneous chain is still valid and complete in the homogeneous limit.

Conclusion and outlook {#conclusion-and-outlook .unnumbered}
======================

In this paper we have shown that the transfer matrix spectrum associated to the most general spin-1/2 representations of the 6-vertex reflection algebras (rational and trigonometric), on general inhomogeneous chains is completely characterized in terms of a second order difference functional equations of Baxter $T$-$Q$ type with an inhomogeneous term depending only on the inhomogeneities of the chain and the boundary parameters. This functional $T$-$Q$ equation has been shown to be equivalent to the SOV complete characterization of the spectrum when the $Q$-functions belong to a well defined set of polynomials. The polynomial character of the $Q$-function is a central feature of our characterization which allows to introduce an equivalent finite system of generalized Bethe ansatz equations. Moreover, we have explicitly proven that our functional characterization holds for all the values of the boundary parameters for which SOV works, clearly identifying the only 3-dimensional hyperplanes in the 6-dimensional space of the boundary parameters where our description cannot be applied. We have also clearly identified the 5-dimensional hyperplanes in the space of the boundary parameters where the spectrum (or a part of the spectrum) can be characterized in terms of a homogeneous $T$-$Q$ equation and the polynomial character of the $Q$-functions is then equivalent to a standard system of Bethe equations. Completeness of this description is a built in feature due to the equivalence to the SOV characterization. The equivalence between our functional $T$-$Q$ equation and the SOV characterization holds for generic values of the $\xi _{a}$ in the $\mathsf{N}$-dimensional space of the inhomogeneity parameters however there exist hyperplanes for which the conditions [(\[xi-conditions\])]{} are not satisfied and so a direct application of the SOV approach is not possible (at least for the separate variables described in [@Fald-KN13]) and the limit of homogeneous chains ($\xi _{a}\rightarrow 0$ $\forall a\in \{1,...,\mathsf{N}\}$) clearly belong to these hyperplanes. From the analyticity of the transfer matrix eigenvalues, of the coefficients of the functional $T$-$Q$ equation and of the inhomogeneous term in it w.r.t. the inhomogeneity parameters it is possible to argue that these functional equations still describes transfer matrix eigenvalues on the hyperplanes where SOV method cannot be applied and, in particular, in the homogeneous limit. However, in all these cases the statements about the simplicity of the transfer matrix spectrum and the completeness of the description by our functional $T$-$Q$ equation are not anymore granted and they require independent proofs. These fundamental issues will be addressed in a future publication. Here we want just to recall that the comparison with the few existing numerical results on the subject seems to suggests that the statement of completeness should be satisfied even in the homogeneous limit of special interest as it allows to reproduce the spectrum of the Hamiltonian of the spin-1/2 open XXZ quantum chains under the most general integrable boundary conditions.

Finally, it is important to note that the form of the Baxter functional equation for the most general spin-1/2 representations of the 6-vertex reflection algebras and in particular the necessity of an inhomogeneous term are mainly imposed by the requirement that the set of solutions is restricted to polynomials. Then the problem to get homogeneous Baxter equations relaxing this last requirement remains an interesting open problem.

Acknowledgements {#acknowledgements .unnumbered}
================

The authors would like to thank E. Sklyanin and V. Terras for discussions. J.M.M. and G. N. are supported by CNRS. N.K and J.M.M. are supported by ANR grant “DIADEMS”. N. K. would like to thank LPTHE, University Paris VI and Laboratoire de Physique, ENS-Lyon for hospitality.

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[^1]: Different methods leading to Bethe ansatz equations have been also proposed under the same boundary conditions by using the framework of the Temperley-Lieb algebra in [@deGP04; @NicRd05] and by making a combined use of coordinate Bethe ansatz and matrix ansatz in [@CraRS10; @CraR12].

[^2]: Some partial results in this direction were achieved in [@FilK11] but only in the special case of double boundary constrains introduced in [YanZ07]{}.

[^3]: See also the papers [@CaoYSW13-1; @CaoYSW13-2; @CaoYSW13-3] for the application of the same method to different models.

[^4]: Here we use notations similar to those introduced in the papers [@CaoJYW2013] and [@Nep-2013] where some inhomogeneous Baxter equation ansatzs appear with the aim to make simpler for the reader a comparison when the limit of homogeneous chain is implemented.
