2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/50678Using the Baxter's T-Q relation derived from the transfer matrix technique, we consider the diagonalization problem of discrete quantum pendulum and discrete quantum sine-Gordon Hamiltonian from the algebraic geometry aspect. For a finite chain system of the size L, when the spectral curve degenerates into rational curves, we have reduced Baxter's T-Q relation into a polynomial equation; the connection of T-Q polynomial equation with the algebraic Bethe Ansatz is clearly established . In particular, for L=4 it is the case of rational spectral curves for the discrete quantum pendulum and discrete sine-Gordon model. To these Baxter's T-Q polynomial equations, we have obtained the complete and explicit solutions with a detailed understanding of their quantitative and qualitative structure. In general the model possesses a spectral curve with a generic parameter. We have conducted certain qualitative study on the algebraic geometry of this high-genus Riemann surface incorporating Baxter's T-Q relation.27 pages, Latex; Some reorganizations and improvement of presentations, and other minor changesHigh Energy Physics - TheoryStatistical MechanicsMathematical PhysicsAlgebraic GeometryExactly Solvable and Integrable Systemstext