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arXiv · 1312.3493

A product formula for the eigenfunctions of a quartic oscillator

Martin Hallnäs·Edwin Langmann
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Abstract

We consider the Schrödinger operator on the real line with an even quartic potential. Our main result is a product formula of the type $ψ_k(x)ψ_k(y) = \int_{\mathbb{R}} ψ_k(z)\mathcal{K}(x,y,z)dz$ for its eigenfunctions $ψ_k$. The kernel function $\mathcal{K}$ is given explicitly in terms of the Airy function $\mathrm{Ai}(x)$, and is positive for appropriate parameter values. As an application, we obtain a particular asymptotic expansion of the eigenfunctions $ψ_k$.

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Martin Hallnäs, Edwin Langmann. 2016-09-12. A product formula for the eigenfunctions of a quartic oscillator. https://doi.org/10.1016/j.jmaa.2015.02.014

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