Exactly solvable models and dynamic quantum systems

Complicated dynamic systems with several degrees of freedom are investigated with the inverse scattering method using an adiabatic approach based on a consistent statement of two adiabatic problems. An algebraic technique based on the parametric inverse problem in an adiabatic representation is developed for reconstructing two-dimensional (time-dependent and time-independent) potentials and the corresponding solutions. The calculated elements of the exchange interaction matrix determine the system of corresponding gauge equations. The main characteristics of the exchange interaction essentially depend on the statement of the parametric inverse problem. Namely, if the parametric problem is specified on the entire axis, then the constraint matrix elements are regular at degeneration points of two levels. The opposite occurs in the case of the radial parametric problem or the parametric problem specified on the semiaxis. The influence of the parametric spectral characteristics of the fast subsystem on the behavior of the slow subsystem is studied. In particular, it is shown that state transitions of a two-level system vanish for a special choice of the normalization functions.