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arXiv · 2606.04635

Gaussian decay for the harmonic oscillator

Manish Chaurasia
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Abstract

We consider the Schr\"odinger equation associated with the harmonic oscillator and show that if the initial data and its Fourier transform are dominated by Gaussian functions of widths $a>0$ and $b>0$, respectively, satisfying $ab<1$, then the evolved solution and its Fourier transform are dominated by a Gaussian of width $\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}- \sqrt{\left(\frac{1}{a}+\frac{1}{b}\right)^2-4}\right),$ for all times except for a discrete set, and for all times in one dimension. In the one-dimensional case, we prove that these estimates are sharp. Moreover, for a more restrictive class of initial data, we establish sharper time-dependent Gaussian bounds.

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Manish Chaurasia. 2026-06-03. Gaussian decay for the harmonic oscillator. https://arxiv.org/abs/2606.04635

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