Calculations of time-dependent observables in non-Hermitian quantum mechanics: The problem and a possible solution

The solutions of the time-independent Schrödinger equation for non-Hermitian (NH) Hamiltonians have been extensively studied and calculated in many different fields of physics by using L² methods that originally have been developed for the calculations of bound states. The existing non-Hermitian formalism breaks down when dealing with wave packets (WPs). An open question is how time-dependent expectation values can be calculated when the Hamiltonian is NH? Using the F-product formalism that was recently proposed by Moiseyev and Lein J. Phys. Chem. 107 7181 (2003) we calculate the time-dependent expectation values of different observable quantities for a simple well-known study test case model Hamiltonian. We carry out a comparison between these results and those obtained from conventional (i.e., Hermitian) quantum mechanics (QM) calculations. The remarkable agreement between these results emphasizes the fact that in NH QM, unlike standard QM, there is no need to split the entire space into two regions, i.e., the interaction region and its surrounding. Our results open a door for a type of WP propagation calculations within the NH QM formalism that until now were impossible. In particular our work is relevant to the many different fields in physics and chemistry where complex absorbing potentials are introduced in order to reduce the propagation calculations to a restricted region in space where the artificial reflections from the edge of the numerical grid or box are avoided.