  In this paper we consider the wave operators $W_{\pm}$ for a Schr\"odinger
operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$
boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential
$V$. The analysis heavily depends on the singularities of the resolvent for
small energy, that is if 0-energy eigenstates exist. If such eigenstates do not
exist $W_{\pm}: L^p \to L^p$ are bounded for $1 \leq p \leq \infty$ otherwise
this is true for $ \frac{n}{n-2} < p < \frac{n}{2} $. The extension to Sobolev
space is discussed.
