Unusual scalar products in Hilbert space of Quantum Mechanics: non-Hermitian square-well model with two coupled channels

Abstract

A pseudo-Hermitian coupled-channel square-well model is proposed, solved and discussed. The domain of parameters is determined where all the bound-state energies (twice degenerate with respect to the second observable which we call "spin") remain real. The standard probabilistic interpretation of the wave functions is achieved by the transition from the original elementary non-physical indefinite pseudo-metric $θ$ to another, more involved but correct positive-definite physical metric $Θ$. In our model the latter step remains comparatively easily feasible. Another fortunate circumstance emerges during the perturbative construction of the metric $Θ$ and, hence, of the correct scalar product and of the physical norm. One finds that at the higher energy levels the influence of all the non-Hermiticities of our model becomes very strongly and progressively suppressed.