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On the metric operator for the imaginary cubic oscillator

P. Siegl and D. Krejčiřík
Phys. Rev. D 86, 121702(R) – Published 4 December 2012

Abstract

We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of nontrivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT-symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.

  • Received 2 May 2012

DOI:https://doi.org/10.1103/PhysRevD.86.121702

© 2012 American Physical Society

Authors & Affiliations

P. Siegl1,2 and D. Krejčiřík2

  • 1Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Avenida Professor Gama Pinto 2, 1649-003 Lisboa, Portugal
  • 2Department of Theoretical Physics, Nuclear Physics Institute ASCR, 25068 Řež, Czech Republic

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Issue

Vol. 86, Iss. 12 — 15 December 2012

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