A non self-adjoint model on a two dimensional noncommutativespace with unbound metric

Fabio Bagarello, Fabio Bagarello, Andreas Fring

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We demonstrate that a non-self-adjoint Hamiltonian of harmonic-oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudobosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT -symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that thesesets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space L2(R2), but instead only D quasibases. As recently proved by one of us, this is sufficient to deduce several interesting consequences.
Original languageEnglish
Number of pages6
JournalPHYSICAL REVIEW A
Volume88
Publication statusPublished - 2013

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operators
Hilbert space
harmonic oscillators
quantum mechanics
eigenvectors
eigenvalues
symmetry

All Science Journal Classification (ASJC) codes

  • Atomic and Molecular Physics, and Optics

Cite this

A non self-adjoint model on a two dimensional noncommutativespace with unbound metric. / Bagarello, Fabio; Bagarello, Fabio; Fring, Andreas.

In: PHYSICAL REVIEW A, Vol. 88, 2013.

Research output: Contribution to journalArticle

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AB - We demonstrate that a non-self-adjoint Hamiltonian of harmonic-oscillator type defined on a two-dimensional noncommutative space can be diagonalized exactly by making use of pseudobosonic operators. The model admits an antilinear symmetry and is of the type studied in the context of PT -symmetric quantum mechanics. Its eigenvalues are computed to be real for the entire range of the coupling constants and the biorthogonal sets of eigenstates for the Hamiltonian and its adjoint are explicitly constructed. We show that despite the fact that thesesets are complete and biorthogonal, they involve an unbounded metric operator and therefore do not constitute (Riesz) bases for the Hilbert space L2(R2), but instead only D quasibases. As recently proved by one of us, this is sufficient to deduce several interesting consequences.

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