On the separation of variables for the modular XXZ magnet and the lattice Sinh-Gordon models

We construct the generalised Eigenfunctions of the entries of the monodromy matrix of the $N$-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $L^2({\mathbb{R}}^N)$. In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of out analysis, we prove the Bystko-Teschner conjecture relative to the structure of the spectrum of the $\boldsymbol{ $\mathtt{\text{B}}$ }(\la)$-operator for the odd length lattice Sinh-Gordon model.