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On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, 115 19, Prague, Czech Republic

    Radek Novák

  2. Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Hlavní 130, 250 68, Řež near Prague, Czech Republic

    Radek Novák

  3. Laboratoire de Mathématiques Jean Leray, 9, Université de Nantes, 2 rue de la Houssinière, 443 22, Nantes Cedex 3, France

    Radek Novák

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Novák, R. On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential. Int J Theor Phys 54, 4142–4153 (2015). https://doi.org/10.1007/s10773-015-2530-5

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