Algebraic Geometry Approach to the Bethe Equation for Hofstadter Type Models

We study the diagonalization problem of certain Hofstadter-type models through the algebraic Bethe ansatz equation by the algebraic geometry method. When the spectral variables lie on a rational curve, we obtain the complete and explicit solutions for models with the rational magnetic flux, and discuss the Bethe equation of their thermodynamic flux limit. The algebraic geometry properties of the Bethe equation on high genus algebraic curves are investigated in cooperationComment: 28 pages, Latex ; Some improvement of presentations, Revised version with minor changes for journal publicatio