Sharp Gaussian decay for the one-dimensional harmonic oscillator

Radchenko, Danylo; Ramos, João P. G.

Type de document :

Pré-publication ou Document de travail

URL permanente :

http://hdl.handle.net/20.500.12210/121991

Titre :

Sharp Gaussian decay for the one-dimensional harmonic oscillator

Auteur(s) :

Radchenko, Danylo [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Ramos, João P. G. [Auteur]
Department of Mathematics [ETH Zurich] [D-MATH]

Date de publication :

2023-05-29

Mot(s)-clé(s) en anglais :

Hermite polynomials
harmonic oscillator
Plancherel-Rotach formula

Discipline(s) HAL :

Mathématiques [math]
Mathématiques [math]/Analyse classique [math.CA]

Résumé en anglais : [en]

We prove a conjecture by Vemuri by proving sharp bounds on $\ell^{\kappa}$ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each $y>0,$ we have \[ \sum_{n \ge ...
Lire la suite >We prove a conjecture by Vemuri by proving sharp bounds on $\ell^{\kappa}$ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each $y>0,$ we have \[ \sum_{n \ge 1} |h_n(x)|^{\kappa} \frac{e^{-\kappa n y}}{n^{\beta}} \ll_y x^{\frac{1}{2} - 2\beta} e^{-\kappa x^2 \tanh(y)/2}, \] for all $x \in \mathbb{R}$ sufficiently large. Our proof involves the classical Plancherel-Rotach asymptotic formula for Hermite polynomials and a careful local analysis near the maximum point of such a bound.Lire moins >

Langue :

Anglais

Commentaire :

5 pages

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

Date de dépôt :

2025-01-24T14:55:08Z

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document
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Proof_of_Vemuri_s_Conjecture.pdf
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2305.18546
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