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Harmonic oscillator perturbed by a decreasing scalar potential

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Abstract

In this paper we study the perturbation \(L=H+V\), where \(H=-\frac{{{d}^{2m}}}{d{{x}^{2m}}}+{{x}^{2m}}\) on \(\mathbb {R}\), \(m\in {{\mathbb {N}}^{*}}\) and V is a decreasing scalar potential. Let \({{\lambda }_{k}}\) be the \(k^{th}\) eigenvalue of H. We suppose that the eigenvalues of L around \({{\lambda }_{k}}\) can be written in the form \({{\lambda }_{k}}+{{\mu }_{k}}\). The main result of the paper is an asymptotic formula for fluctuation \(\{ {{\mu }_{k}} \}\) which is given by a transformation of V. In the case \(m=1\) we recover a result on the harmonic oscillator.

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Acknowledgements

Many thanks to our -pert for developing this class file.

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Correspondence to Mohamed Ali Tagmouti.

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Aarab, I., Tagmouti, M.A. Harmonic oscillator perturbed by a decreasing scalar potential. J. Pseudo-Differ. Oper. Appl. 11, 141–157 (2020). https://doi.org/10.1007/s11868-019-00284-4

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Keywords

  • Averaging method
  • Pseudo-differential operator
  • Perturbation theory
  • Spectrum
  • Eigenvalue asymptotics

Mathematics Subject Classification

  • Primary 99Z99
  • Secondary 00A00