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 $L^1\to L^\infty$ dispersive estimates associated to the evolution operator $e^{-i t H_\alpha}$. 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 $e^{-i t H_\alpha}$ depends on $\alpha$. More precisely we show that the decay rate of $e^{-i t H_\alpha}$ for $t\to\infty$ can be made arbitrarily large provided we choose $\alpha$ large enough and consider a suitable operator norm.