Competing models for the relaxation of tunneling and paramagnetic defects

The phonon-assisted tunneling problem (which to a large degree is isomorphic to the phonon-assisted paramagnetic relaxation problem) is handled by means of unitary exponential transformations.In view of the hierarchical possibilities for the 3 Hamiltonian constituents (phonons, bare tunneling, coupling) there are 4 fundamental exponential transformations. These are discussed. Further, an exactly solvable model is studied. Finally, the underlying Fano problem is investigated by means of an exactly solvable model. It is found that a) the reduction of the tunneling frequency is modified by those lattice modes which lie below the bare tunneling splitting (there even may be an increase of); b) there may be more than a single relaxation path; c) there may be a relaxation process which approximately does not depend on at all; in this mechanism the tunneling behaviour is transferred to the phonon system.