From self-adjoint to non self-adjoint harmonic oscillators: physical consequences and mathematical pitfalls

Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the Hamiltonian H changes drastically the related functional structure. In particular, we show that even a small deviation from strict self-adjointness of H produces two deep consequences, not well understood in the literature: First of all, the original orthonormal basis of H splits into two families of biorthogonal vectors. These two families are complete but, contrarily to what often claimed for similar systems, none of them is a basis for the Hilbert space H. Second, the so-called metric operator is unbounded, as well as its inverse. In the second part of the paper, after an extension of some previous results on the so-called D pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.