Spectral properties of higher order anharmonic oscillators

We discuss spectral properties of the self-adjoint operator $$-d^2/dt^2 + (t^{k+1}/(k+1)-\alpha)^2$$ 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, Helffer-Morame, Aramaki, Helffer-Kordyukov and Helffer.