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Journal of Mathematical Sciences

, Volume 165, Issue 1, pp 110–126 | Cite as

Spectral properties of higher order anharmonic oscillators

  • B. Helffer
  • M. Persson
Article

We discuss spectral properties of the selfadjoint operator \( \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} \) in L 2(ℝ) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schrödinger operators with magnetic field. Bibliography: 13 titles. Illustrations: 2 figures.

Keywords

Spectral Property Ground State Energy Trial Function Schwarz Inequality Selfadjoint Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Département de Mathématiques Bâtiment 425Univ Paris-Sud et CNRSOrsayFrance
  2. 2.Department of Mathematical Sciences Aarhus UniversityAarhus CDenmark

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