Sharp Gaussian decay for the one-dimensional harmonic oscillator

2023/05/29 by Danylo Radchenko, João P. G. Ramos, Radchenko, Danylo +1 · 3 citations
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2305.18546

openalex publication_date 2023/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove a conjecture by Vemuri by proving sharp bounds on ℓ^κ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each y>0, we have ∑_{n ≥ 1} |h_n(x)|^κ \frace^{-κn y}n^β ≪_y x^{(1)/(2) - 2β} e^{-κx2 \tanh(y)/2}, for all x ∈ 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.

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• Dispersive equations with invariant measures
• On the Carlitz-Mehler formula for Hermite polynomials
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