Asymptotically exact wave functions of the harper equation

We present asymptotically exact wave functions of an incommensurate Harper equation-one-dimensional Schrödinger equation of one particle on a lattice in a cosine potential. The wave functions can be written as an infinite product of string polynomials. The roots of these polynomials are solutions of Bethe equations. They are classified according to the string hypothesis. The string hypothesis gives asymptotically exact values of roots and reveals the hierarchical structure of the spectrum of the Harper equation.