### Workshop Nosevol #3

Mikael Persson :
Uniqueness of minima of anharmonic oscillators.

In spectral theory of magnetic Schrödinger operators, some
families of Sturm--Liouville operators often appear. For $k\geq 1$ we
study the parameter-dependent family of self-adjoint operators
$H(\alpha)u(t) = -u''(t)+(t^(k+1)/(k+1)-\alpha)^2u(t)$ in
$L^2(\mathbb{R})$. Our aim is to show that the mapping
$\mathbb{R}\ni\alpha\mapsto \lambda_1(\alpha)$ (where
$\lambda_1(\alpha)$ denotes the smallest eigenvalue of $H(\alpha)$
possess a unique non-degenerate minimum. We will also discuss the
connection between these operators and magnetic Schrödinger operators.
This is joint work with B. Helffer and S. Fournais.