A uniqueness theorem for higher order anharmonic oscillators

Abstract

We study for $\alpha \in \mathbb{R}$,$k\in \mathbb{N}\setminus \{0\}$ the family of self-adjoint operators

in $L^2(\mathbb{R})$ and show that if $k$ is even then $\alpha =0$ gives the unique minimum of the lowest eigenvalue of this family of operators. Combined with earlier results this gives that for any $k\ge 1$, the lowest eigenvalue has a unique minimum as a function of $\alpha$.