On Hermitian and non-Hermitian flux conservation for quantum tunneling decay

We consider exact expansions of the decaying wave solution to the Schrödinger equation in terms of continuum energy solutions (Hermitian) and resonant states involving transient functions (non-Hermitian) discussed in Ref. (G. García-Calderón et al. Phys Scr T151(T151): 014076 https://doi.org/10.1088/0031-8949/2012/T151/014076, 2012), to analyze the conditions that guarantee flux conservation in quantum tunneling decay. For the Hermitian case, we find that flux conservation depends on the difference of two arbitrary energies of the continuum which establishes an intriguing correlation with the continuum wave solutions whereas for the non-Hermitian case, it provides a condition that relates the imaginary part of each complex pole of the outgoing Green’s function of the problem with the corresponding resonant state.