Geometric amplitude, adiabatic invariants, quantization, and strong stability of Hamiltonian systems

Considered is a linear set of ordinary differential equations with a matrix depending on a set of adiabatically varying parameters. Asymptotic solutions have been constructed. As has been shown, an important characteristic determining the qualitative portrait of the system is the real part of Berry’s complex geometric phase, which we call the geometric amplitude. For systems with purely imaginary eigenvalues the equivalence has been proven in the adiabatic approximation of system sets without geometric amplitude, Hamiltonian systems, quantizable systems, and strongly stable systems. Classification of the systems without geometric amplitude is given with respect to the kind of matrix of the initial system.