SHARP GAUSSIAN DECAY FOR THE ONE-DIMENSIONAL HARMONIC OSCILLATOR

We prove a conjecture by Vemuri [Hermite expansions and Hardy's theorem, arXiv:0801.2234, 2008] by proving sharp bounds on l(kappa) sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each y > 0, we have Sigma( n >= 1) |hn(x)|(kappa)e -kappa ny /n(beta)< y x(1)- kappa/2 -2 beta(e)-kappa x(2 tanh(y)/2),for all x is an element of 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.