Schrödinger operators on a half-line with inverse square potentials

Abstract. We consider Schrödinger operators Hα given by equation (1.1) below. We study the asymptotic behavior of the spectral density E(Hα, λ) for λ → 0 and the L1 → L ∞ dispersive estimates associated to the evolution operator e−itHα. In particular we prove that for positive values of α, the spectral density E(Hα, λ) tends to zero as λ → 0 with higher speed compared to the spectral density of Schrödinger operators with a short-range potential V. We then show how the long time behavior of e−itHα depends on α. More precisely we show that the decay rate of e−itHα for t→ ∞ can be made arbitrarily large provided we choose α large enough and consider a suitable operator norm