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 \[ -\frac{d^2}{dt^2}+\Bigl(\frac{t^{k+1}}{k+1}-α\Bigr)^2 \] 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 $k\ge 1$, the lowest eigenvalue has a unique minimum as a function of $\alpha$.

arXiv comment10 pages, 2 figures, (updated with bibliography included)

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• Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells

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• Little Magnetic Book