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 $\mathcal{H}$. 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 $\mathcal{D}$ pseudobosons, we discuss some aspects of our extended harmonic oscillator from this different point of view.

• Received 2 September 2013

DOI:https://doi.org/10.1103/PhysRevA.88.032120