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

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.