Sharp Gaussian decay for the one-dimensional harmonic oscillator

Abstract

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 \[ \sum_{n \ge 1} |h_n(x)|^κ \frac{e^{-κn y}}{n^β} \ll_y x^{\frac{1}{2} - 2β} e^{-κx^2 \tanh(y)/2}, \] 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.

5 pages