2007, Vol.14, pp.217-227

Complex systems, such as nuclear, atomic, molecular ones and so on, are characterized by some degrees of freedom, separation of which in strongly interacting systems, as a rule, are not valid. One of the most useful methods in the treatment of the quantum dynamical systems with some degrees of freedom is the adiabatic representation method.
Here we investigate the role played by level crossing for collective motion in the presence of "fast" dynamics of separate particles in the system within the adiabatic representation. The method presented permits one to construct a wide class of potentials and corresponding solutions of the parametric equation in a closed analytical form and, after that, to calculate the matrix elements of the exchange interaction. It was shown that the main features of the exchange interaction determining the "slow" subsystem Hamiltonian essentially depend on the character of the parametric Hamiltonian: namely, it is given on the semi-axis or on the entire axis. As a consequence, the problems of level crossing are different in both cases. It was established that in the case of the parametric problem on the entire axis the induced scalar and vector potentials and the basis functions are not singular at the degeneracy points of the two states, while in the parametric problem on the half-axis, the potential, together with its eigenfunctions and matrix elements of the exchange interaction, are singular at these points. In particular, we have found that in the parametric problem on the entire axis for a special choice of the normalization functions, the potential is transparent and symmetric in the "fast" variable and the exchange interaction between the bound states for two-level systems are equal to zero for all values of the "slow" variables, even at the point of the degeneracy.
Key words: Inverse scattering problem, level crossing, exactly solvable models

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