### Abstract

Original language | English |
---|---|

Number of pages | 6 |

Journal | PHYSICAL REVIEW A |

Volume | 88 |

Publication status | Published - 2013 |

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### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics

### Cite this

*PHYSICAL REVIEW A*,

*88*.

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

Research output: Contribution to journal › Article

*PHYSICAL REVIEW A*, vol. 88.

}

TY - JOUR

T1 - A non self-adjoint model on a two dimensional noncommutativespace with unbound metric

AU - Bagarello, Fabio

AU - Bagarello, Fabio

AU - Fring, Andreas

PY - 2013

Y1 - 2013

N2 - 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.

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.

UR - http://hdl.handle.net/10447/95773

M3 - Article

VL - 88

JO - PHYSICAL REVIEW A

JF - PHYSICAL REVIEW A

SN - 1050-2947

ER -