**Abstract: ** The Schr\"odinger operator $H=-\Delta+V$, where $-\Delta$ is the
Laplace--Beltrami operator on a complete connected Riemannian
manifold $X$ of dimension $\nu$ is considered. We suppose that $X$
is a manifold of bounded geometry and $V$ satisfies the
conditions: (1) $V_+\in L^p_{\rm loc}(X)$, where $p=2$ if
$\nu\le3$ and $p>\nu/2$ if $\nu\ge4$; (2) $V_-\in L^p(X)$, where
$p=2$ if $\nu\le2$ and $p>\nu-1$ if $\nu\ge3$. The continuity of
the Green function of $H$ outside the diagonal and the continuity
of the heat kernel for $H$ are proven. The estimates of norms of
the propagator and of the resolvent in the $L^p$-scale are given.
The continuity of the eigenfunctions and of the kernels for the
spectral projections is discussed. The results are obtained
jointly with J.~Br\"uning and K.~Pankrashkin.