Bernard Helffer, Mikael Persson
Spectral Properties of Higher Order Anharmonic Oscillators
Preprint series: ESI preprints

MSC:

47A75 Eigenvalue problems, See also {49Rxx}

47E05 Ordinary differential operators, See also {34Bxx, 34Lxx,

34L15 Estimation of eigenvalues, upper and lower bounds

34B05 Linear equations

Abstract: We discuss spectral properties of the self-adjoint operator
\begin{equation*}
-\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-\alpha\Bigr)^2
\end{equation*}
in $L^2(\mathbb{R})$ for odd integers $k$. We prove that the minimum over
$\alpha$ of the ground state energy of this operator is attained at a unique
point which tends to zero as $k$ tends to infinity. Moreover, we show that
the minimum is non-degenerate.
These questions arise naturally in the spectral analysis of Schr\"{o}dinger
operators with magnetic field.
This extends or clarifies previous results
by Pan-Kwek~\cite{pakw}, Helffer-Morame~\cite{hemo1}, Aramaki~\cite{ara},
Helffer-Kordyukov~\cite{heko1,heko5,heko4} and Helffer~\cite{helf}.


Keywords: eigenvalue estimation, anharmonic oscillator, spectral parameter