### Workshop Nosevol #3

Françoise Truc :
Schrödinger operators on a half-line with inverse square potentials.

We consider Schr\"odinger operators $H_\alpha=-\frac{d^2}{dx^2} +
\frac{\alpha}{x^2}\$ on the half-line,
with Dirichlet condition at $x=0$.
We study the asymptotic behavior of the spectral density $E(H_\alpha,
\lambda)$ for $\lambda \to 0$ 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 that the decay rate of the evolution operator can be made
arbitrarily large provided we choose $\alpha$ large enough and
consider a suitable operator norm.
This is a joint work with Hynek Kovarik (University of Brescia).