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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.