**Abstract: **
We consider the Schroedinger operator on a Riemannian manifold $A$
with a potential that localizes certain states close to a submanifold $C$.
When scaling the potential in the directions normal to $C$ by a parameter
$\varepsilon\ll 1$, the solutions concentrate in an
$\varepsilon$-neighborhood of $C$. This situation occurs for example in
quantum wave guides and for the motion of nuclei in electronic potential
surfaces in quantum molecular dynamics. We derive an effective Schroedinger
operator on the submanifold $C$ that approximates the spectrum and the
dynamics of the full operator on $A$ up to terms of order $\varepsilon^3$.