Using the framework of the quantum separation of variables (SoV) for higher rank quantum integrable lattice models [1], we introduce some foundations to go beyond the obtained complete transfer matrix spectrum description, and open the way to the computation of matrix elements of local operators. This first amounts to obtain simple expressions for scalar products of the so-called separate states (transfer matrix eigenstates or some simple generalization of them). In the higher rank case, left and right SoV bases are expected to be pseudo-orthogonal, that is for a given SoV co-vector, there could be more than one non-vanishing overlap with the vectors of the chosen right SoV basis. For simplicity, we describe our method to get these pseudo-orthogonality overlaps in the fundamental representations of the $\mathcal{Y}(gl_3)$ lattice model with $N$ sites, a case of rank 2. The non-zero couplings between the co-vector and vector SoV bases are exactly characterized. While the corresponding SoV-measure stays reasonably simple and of possible practical use, we address the problem of constructing left and right SoV bases which do satisfy standard orthogonality. In our approach, the SoV bases are constructed by using families of conserved charges. This gives us a large freedom in the SoV bases construction, and allows us to look for the choice of a family of conserved charges which leads to orthogonal co-vector/vector SoV bases. We first define such a choice in the case of twist matrices having simple spectrum and zero determinant. Then, we generalize the associated family of conserved charges and orthogonal SoV bases to generic simple spectrum and invertible twist matrices. Under this choice of conserved charges, and of the associated orthogonal SoV bases, the scalar products of separate states simplify considerably and take a form similar to the $\mathcal{Y}(gl_2)$ rank one case.
The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
Univ Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique（里昂大学、巴黎高等师范学院、克洛德·贝尔纳大学、法国国家科学研究中心、物理实验室）；Université Paris-Saclay, CNRS, LPTMS（巴黎萨克雷大学、法国国家科学研究中心、理论与数学物理实验室）
An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. I: D'Alembert-Kirchhoff-like formulae
Holomorphic bundles and Scalar Difference Operators: One-Point Constructions
有效地构造了秩l>1的一维差分算子的交换环及其变形。我们的分析结构是基于所谓的'' Tyurin参数''的稳定框架全纯向量丛代数曲线的亏格等于g和陈数等于lg。这些参数大量使用本作者已经在1978年至1980年的微分算子。在离散情况下，它们的变形由2D Toda Lattice hierarhy而不是KP控制。在l=2，g=1的情况下，这里出现了新的可积系统。高秩差分算子的理论比以前文献中仅使用谱曲线上的2点构造（即“无限点”的数目等于2）的秩1情形丰富得多。对于每个偶数秩l=2k，一点构造出现在这个问题中。只有在这种情况下，交换环依赖于功能参数。两点构造将在接下来的工作中研究：即使对于更高的秩l>1，这种情况也可以在Theta函数中解决。对于秩l>1的单点结构则不是这样。
An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. II: Resolvents formulae
