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

Fabio Bagarello, Bagarello

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)1-5
Number of pages5
JournalPHYSICAL REVIEW A
Volume88
Publication statusPublished - 2013

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

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