Exact hierarchical solution of the damped oscillator problem

As a alternative to other exact approaches we present a procedure, in which the eigenvectors of a coupled oscillatory system are calculated in a hierarchically exact (N) manner. In the system a single oscillator is coupled to a dense sequence of background oscillators, which among themselves are not mutually coupled (Oscillatory Fano problem). The eigenvectors beeing known, all decay laws become computable. Two of them are given. It is strongly emphasized thatquite different decay laws evolve fordifferent initial conditions.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday