A uniqueness theorem for higher order anharmonic oscillators

We study for $α\in\R$, $k \in {\mathbb N} \setminus \{0\}$ the family of self-adjoint operators $-\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-α\Bigr)^2$ in $L^2(\R)$ and show that if $k$ is even then $α=0$ 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 $α$.

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