Sharp Gaussian decay for the one-dimensional harmonic oscillator

We prove a conjecture by Vemuri by proving sharp bounds on ${\mathrm{\ell }}^{\kappa }$ sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each $y>0,$ we have $$\sum _{n\ge 1}\mathrm{\mid }{h}_n(x){\mathrm{\mid }}^{\kappa }\frac{e^{-\kappa ny}}{n^{\beta }}{\ll }_y{x}^{\frac{1}{2}-2\beta }{e}^{-\kappa x^2\tanh (y)\mathrm{/}2},$$ 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.