# A uniqueness theorem for higher order anharmonic oscillators

### Søren Fournais

University of Aarhus, Denmark### Mikael Persson Sundqvist

Lund University, Sweden

## Abstract

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

$-\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-\alpha\Bigr)^2$

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 $k \geq 1$, the lowest eigenvalue has a unique minimum as a function of $\alpha$.

## Cite this article

Søren Fournais, Mikael Persson Sundqvist, A uniqueness theorem for higher order anharmonic oscillators. J. Spectr. Theory 5 (2015), no. 2, pp. 235–249

DOI 10.4171/JST/96