# Spectral properties of higher order anharmonic oscillators

- First Online:

- Received:

DOI: 10.1007/s10958-010-9784-5

- Cite this article as:
- Helffer, B. & Persson, M. J Math Sci (2010) 165: 110. doi:10.1007/s10958-010-9784-5

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.