Abstract
We extend a randomization method, introduced by Shiffman–Zelditch and developed by Burq–Lebeau on compact manifolds for the Laplace operator, to the case of \({\mathbb{R}^d}\) with the harmonic oscillator. We construct measures, thanks to probability laws which satisfies the concentration of measure property, on the support of which we prove optimal-weighted Sobolev estimates on \({\mathbb{R}^d}\). This construction relies on accurate estimates on the spectral function in a non-compact configuration space. As an application, we show that there exists a basis of Hermite functions with good decay properties in \({L^{\infty}(\mathbb{R}^{d})}\), when d ≥ 2.
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Communicated by Jan Derezinski.
D. R. was partly supported by the grant “NOSEVOL” ANR-2011-BS01019 01.
L.T. was partly supported by the grant “HANDDY” ANR-10-JCJC 0109.
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Poiret, A., Robert, D. & Thomann, L. Random-Weighted Sobolev Inequalities on \({\mathbb{R}^d }\) and Application to Hermite Functions. Ann. Henri Poincaré 16, 651–689 (2015). https://doi.org/10.1007/s00023-014-0317-5
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DOI: https://doi.org/10.1007/s00023-014-0317-5