On spectral equations for an evolution operator of a $q$-oscillator lattice

TL;DR

This work addresses the spectrum of a relativistic evolution operator $\boldsymbol{U}$ for a two-dimensional $q$-oscillator Kagomé lattice, constructed from the tetrahedron equation. It develops a coordinate Bethe-Ansatz–type approach to formulate algebraic spectral equations that define eigenvalues and eigenstates, and it uncovers a structured, XXZ-like organization of these equations in terms of symmetric polynomials $P_{n,N}(q)$. The key contribution is a conjectured general form of the spectral equations, including a generating function $\mathcal{P}_{N}(z;q)$ and a rectangular-sublattice case giving $\mathcal{P}_{K\times L}(z;q)=(-q^{1-L}z;q^2)_L^K$, together with symmetry properties and $q\to 1$ limits. This provides a principled, though partially verified, pathway to Liouville-integrable spectra in 3D quantum lattice systems and offers connections to known Bethe-Ansatz structures in related models.

Abstract

We propose a set of algebraic equations describing eigenvalues and eigenstates of a relativistic evolution operator for a two-dimensional $q$-oscillator Kagomé lattice. Evolution operator is constructed with the help of $q$-oscillator solution of the Tetrahedron Equation. We focus on the unitary regime of the evolution operator, so our results are related to 3d integrable systems of the quantum mechanics. Our conjecture is based on a two-dimensional lattice version of the coordinate Bethe-Ansatz.