Irreversibility in the Framework of Hermitian and Non-Hermitian Treatments of Resonance States

Abstract

Quasilocalized wavepackets that are formed momentarily in the continuous spectrum of real systems give rise to resonance states that affect the time evolution from the stationary states at t = ?∞ to the stationary states at t = +∞. Provided the excitation process is excluded from consideration, it is valid to treat this class of states as nonstationary, formed at t = 0 and decaying into the continuum of free particle states. The singularity of the solution of the time-dependent Schrodinger equation at t = 0 and the cut of the sectionally analytic resolvent operator on the real energy axis render the system non-Hermitian, with two solutions defined in terms of adjoint spaces and satisfying two complex eigenvalue Schrodinger equations, one for t > 0 and one for t < 0. Accordingly, the energy distribution driving the decay for t = 0 is complex, defined by the Green’s function of the system, and not real, defined by the density g(E). The why, the how and the consequences as regards the understanding and the computation of such states are discussed in this paper. The theoretical frameworks which are used are quantum mechanics plus appropriate physical constraints as well as semiclassical mechanics via path integrals.