On the theory of resonant scattering

A theory is presented which describes the time development as well as the stationary properties of a quantum-mechanical system with unstable discrete states. The formalism enables one to compare and extend the existing resonance theories, especially those of Fano (1961) and Feshbach (1952-67). An important new condition is derived, which determines the character of the time development. If this condition is not satisfied a non-exponential and also non-oscillatory time-decay in a many-resonance system may be observed. This behaviour appears to be connected with a time delay which is predicted to be pulsed at equal intervals in the case of multi-channel scattering and a cross section which can be anomalously narrow. The new condition, which is a commutator relation between two matrices appears to be connected with the so-called `overlap' introduced by Mies (1969). An expression is also presented for the auto-ionization in the case of many resonances, which is a generalization of the well-known result of Fano for one resonance