, Volume 165, Issue 1, pp 110-126
Date: 26 Jan 2010

Spectral properties of higher order anharmonic oscillators

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We discuss spectral properties of the selfadjoint operator \( \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \\ \\ \end{gathered} \) 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.

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