Implications of losing Hermiticity in quantum mechanics

In this note, we revisit a system of one self-interacting boson, described by a non-Hermitian Hamiltonian H acting on an infinite dimensional Hilbert space H. We determine the eigenfunctions of the Hamiltonian and of its adjoint, which constitute complete biorthogonal sets. The probabilistic interpretation of quantum mechanics is not compatible with the metric inherited from H, and attempts to overcome this problem are presented. Consequences of losing self-adjointness in the quantum mechanical context are discussed and the necessity of a careful mathematical analysis of unbounded operators is emphasized.