An example of phase holonomy in WKB theory

This paper discusses the application of WKB theory to Harper's equation $"+, + $n-, +2a cos(2.rrpn +SI$, = E$", in the case in which p is very close to a rational number, p / q. The W K B wavefunction fo r this system is a vector valued quantity, proportional to an eigenvector U of a matrix H ( x, p) , which is parametrised by the phase space coordinates x and p. The complex phase of U is determined by a non-holonomic connection rule; when transported around a cycle and in phase space, U is multiplied by a phase factor ely(. This phase change manifests itself as a modification of the Bohr-Sommerfeld quantisa-tion condition. 1.