PT-symmetric operators and metastable states of the 1D relativistic oscillators

  title={PT-symmetric operators and metastable states of the 1D relativistic oscillators},
  author={Riccardo Giachetti and Vincenzo Grecchi},
  journal={arXiv: Mathematical Physics},
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second order separated operators giving the resonances of the problem by complex dilation. The same operators have unique extensions as closed PT-symmetric operators defining infinite positive energy levels converging to the Schroedinger ones as c tends to infinity. Such energy levels and their eigenfunctions give directly a definite choice of metastable states of the problem. Precise numerical… 

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  • Rev. A 80, 032107
  • 2009
  • Math. Phys. 104, 229 (3), 543-564
  • 2002
  • London Math. Soc. 3, 159 and 169
  • 1961
  • Rev. Lett. 101, 190401
  • 2008
A: Math
  • Gen. 26, 5541
  • 1993
  • Lett. A 374, 1021
  • 2010
A: Math
  • Gen. 42, 355302
  • 2009
  • Lett. 101, 230404
  • 2008