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

TL;DR

This paper constructs eigenfunctions for the modular XXZ magnet and lattice Sinh-Gordon models, proving their completeness and resolving a conjecture about the spectrum of a key operator.

Contribution

It introduces a new technique for proving completeness of eigenfunctions and confirms the Bystko-Teschner conjecture for the spectrum of the B-operator in the lattice Sinh-Gordon model.

Findings

Eigenfunctions form a complete orthogonal system in L^2(R^N)

Developed a new technique for proving completeness

Confirmed the Bystko-Teschner conjecture for the spectrum

Abstract

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 $\boldsymbol{ \texttt{B} }(\la)$-operator for the odd length lattice Sinh-Gordon model.