Journal of Spectral Theory

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Volume 5, Issue 2, 2015, pp. 235–249
DOI: 10.4171/JST/96

Published online: 2015-06-02

A uniqueness theorem for higher order anharmonic oscillators

Søren Fournais[1] and Mikael Persson Sundqvist[2]

(1) University of Aarhus, Denmark
(2) Lund University, Sweden

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 $\alpha$.

Keywords: Eigenvalue estimation, anharmonic oscillator, spectral parameter

Fournais Søren, Persson Sundqvist Mikael: A uniqueness theorem for higher order anharmonic oscillators. J. Spectr. Theory 5 (2015), 235-249. doi: 10.4171/JST/96