@UNPUBLISHED{publication1, title = "Non self-adjoint operators with real spectra and extensions of quantum mechanics", author = "{BEBIANO, Nat{\'{a}}lia} and {PROVID{\^{E}}NCIA, Jo{\~{a}}o da}", year = "2018-12-12", abstract = "In this article, we review the quantum mechanical setting associated to a non self-adjoint Hamiltonian with a real spectrum. Spectral properties of the Hamiltonian of a Swanson-like model are investigated. The eigenfunctions associated to the real simple eigenvalues are shown to form complete systems but not a (Riesz) basis, which gives rise to difficulties in the rigorous mathematical formulation of quantum mechanics. A new inner product, appropriate for the physical interpretation of the model, has been consistently introduced. The dynamics of the system is described. Some specificities of the theory of non self-adjoint operators with implications in quantum mechanics are discussed. ", url = "http://www.mat.uc.pt/preprints/eng\_2018.html", note = "" }