International Journal of Theoretical Physics

, Volume 54, Issue 11, pp 4142–4153 | Cite as

On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

  • Radek Novák


We study the Schrödinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small perturbations and the operator cannot be similar to a self-adjoint operator via a bounded and boundedly invertible transformation. It is shown that its eigenfunctions form a complete set in the Hilbert space of square-integrable functions; however, they do not form a Schauder basis.


Pseudospectrum Harmonic oscillator Imaginary qubic potential 𝓟𝓣-symmetry Semiclassical method 



The research was supported by the Czech Science Foundation within the project 14-06818S and by Grant Agency of the Czech Technical University in Prague, grant No. SGS13/217/OHK4/3T/14. The author would like to express his gratitude to David Krejčiřík, Petr Siegl, Joseph Viola and Miloš Tater for valuable discussions.


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePragueCzech Republic
  2. 2.Department of Theoretical Physics, Nuclear Physics InstituteAcademy of Sciences of the Czech RepublicŘež near PragueCzech Republic
  3. 3.Laboratoire de Mathématiques Jean Leray, 9Université de NantesNantes Cedex 3France

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