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Spectral properties of higher order anharmonic oscillators

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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.

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Correspondence to B. Helffer.

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Translated from Problems in Mathematical Analysis 44 January, 2010, pp. 99–114.

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Helffer, B., Persson, M. Spectral properties of higher order anharmonic oscillators. J Math Sci 165, 110–126 (2010). https://doi.org/10.1007/s10958-010-9784-5

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