Hermitian and non-Hermitian formulations of the time evolution of quantum decay

Abstract

This work discusses Hermitian and non-Hermitian formulations for the time evolution of quantum decay, that involve respectively, continuum wave functions and resonant states, to show that they lead to an identical description for a large class of well behaved potentials. Our approach is based on the analytical properties of the outgoing Green's function to the problem in the complex wave number plane.

Analytical study of quadratic and nonquadratic short-time behavior of quantum decay

“…The resonant states u n (x) satisfy the Schrödinger equation of the problem obeying purely outgoing boundary conditions and hence they also include the bound and antibound states of the problem. It is worth stressing that the resonant state formulation yields exactly the same results as a calculation using continuum states [24].…”

Time-domain resonances and the ultimate fate of a decaying quantum state

“…Although the resonant-state approach is non-Hermitian, it yields exactly the same results as a Hermitian description based on the continuum wave functions to the problem [27,30]. The essential difference, and also the main advantage of the resonant-state approach, is that the non-Hermitian formulation provides explicit analytical expressions for both exponential and nonexponential contributions to decay, whereas the Hermitian description corresponds to a black-box type of calculation that requires numerical integration over the momenta at every instant of time and hence it is difficult to foresee its behavior as a function of time.…”

Underlying non‐Hermitian character of the Born rule in open quantum systems

“…In fact, the present work has been motivated by the recent result that the time evolution of decay by tunneling involving continuum wave functions yields identical results to that using resonance (quasinormal) states . The reason is that both basis follow from the analytical properties of the Green's function to the problem.…”

“…In order to illustrate our findings, we consider the exactly solvable model given by a δ‐shell potential of intensity λ and radius a , for zero angular momentum, and an initial state, the infinite box state, This model has also been used in Ref. [] to illustrate that the formulations or the probability density in terms of continuum wave functions and resonance (quasinormal) states yield identical results for the time evolution of decay.…”