We introduce fractional monodromy in order to characterize certain non-isolated critical values of the energy-momentum map of integrable Hamiltonian dynamical systems represented by nonlinear resonant two-dimensional oscillators. We give the formal mathematical definition of fractional monodromy, which is a generalization of the definition of monodromy used by other authors before. We prove that the 1:(−2) resonant oscillator system has monodromy matrix with half-integer coefficients and discuss manifestations of this monodromy in quantum systems.

We s h ow t hat t he h ydrogen atom in orthogonal electric and m agnetic elds has a special property of certain integrable classical Hamiltonian systems known as monodromy. T h e strength o f t he elds is assumed to b e s m all enough to v alidate t he u s e o f a normal form H snf which i s o b t ained from a two s t ep normalization of the o r i g i n al system. We c o n s i d er the l e v el sets o f H snf on the second r e d uced phase space. For an open set of eld parameters we s h ow t hat t here is a special dynamically invariant s e t which is a \doubly pinched 2-torus". This implies that t he i n tegrable Hamiltonian H snf has monodromy. Manifestation of monodromy in quantum m echanics is also discussed.

No abstract