Schr\"odinger operators on a half-line with inverse square potentials

We consider Schr^odinger operators $H_{\alpha }$ given by equation (1.1) below. We study the asymptotic behavior of the spectral density $E(H_{\alpha },\lambda )$ when $\lambda$ goes to $0$ and the $L^1\to L^{\mathrm{\infty }}$ dispersive estimates associated to the evolution operator $e^{-it{H}_{\alpha }}$. In particular we prove that for positive values of $\alpha$, the spectral density tends to zero as $\lambda \to 0$ with higher speed compared to the spectral density of Schr"odinger operators with a short-range potential $V$. We then show how the long time behavior of $e^{-it{H}_{\alpha }}$ depends on $\alpha$. More precisely we show that the decay rate of $e^{-it{H}_{\alpha }}$ for $t\to \mathrm{\infty }$ can be made arbitrarily large provided we choose $\alpha$ large enough and consider a suitable operator norm.