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

TL;DR

This paper introduces algebraic spectral equations for a relativistic evolution operator in a 2D q-oscillator Kagome lattice, connecting 3D integrable systems with a lattice Bethe-Ansatz approach.

Contribution

It develops a set of algebraic equations for eigenvalues and eigenstates of a 2D q-oscillator lattice evolution operator using a 3D integrability framework.

Findings

Derived algebraic spectral equations for the lattice operator.

Linked the spectral problem to 3D integrable quantum systems.

Proposed a lattice Bethe-Ansatz for the spectral analysis.

Abstract

We propose a set of algebraic equations describing eigenvalues and eigenstates of a relativistic evolution operator for a two-dimensional $q$-oscillator Kagom\'e 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.