Generalization of Berry’s geometric phase, equivalence of the Hamiltonian nature, quantizability and strong stability of linear oscillatory systems, and conservation of adiabatic invariants

Abstract

A linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters is considered. Its asymptotic solutions are constructed to an arbitrary accuracy in the adiabaticity parameter ε. By extending the phase space of the system not only with the space of parameters, like in the theory of Berry’s phase, but also with the space of their derivatives, it proves possible to represent the phase of the solution as a sum of the dynamic phase and a generalized geometric phase (determined by a contour integral in the space of parameters and derivatives). Hence, it is possible to obtain the asymptotic results to any degree of accuracy in ε, while earlier they were obtainable only in the first-order adiabatic approximation. Namely, for linear oscillatory adiabatic systems, the quantizability, parametric stability, and the Hamiltonian nature of the system are equivalent properties. As a consequence, one can obtain the well-known result regarding conservation with accuracy to any power of ε of adiabatic invariants in a Hamiltonian system on a torus. An important point is that the generalized geometric phase can appear, in contrast to Berry’s phase, with only one real varying parameter.