  We consider the time-dependent Schr\"odinger equation on a Riemannian
manifold $\mathcal{A}$ with a potential that localizes a certain class of
states close to a fixed submanifold $\mathcal{C}$. When we scale the potential
in the directions normal to $\mathcal{C}$ by a parameter $\varepsilon\ll 1$,
the solutions concentrate in an $\veps$-neighborhood of $\mathcal{C}$. We
derive an effective Schr\"odinger equation on the submanifold $\mathcal{C}$ and
show that its solutions, suitably lifted to $\mathcal{A}$, approximate the
solutions of the original equation on $\mathcal{A}$ up to errors of order
$\varepsilon^3|t|$ at time $t$. Furthermore, we prove that the eigenvalues of
the corresponding effective Hamiltonian below a certain energy coincide up to
errors of order $\varepsilon^3$ with those of the full Hamiltonian under
reasonable conditions.
  Our results hold in the situation where tangential and normal energies are of
the same order, and where exchange between these energies occurs. In earlier
results tangential energies were assumed to be small compared to normal
energies, and rather restrictive assumptions were needed, to ensure that the
separation of energies is maintained during the time evolution. Most
importantly, we can allow for constraining potentials that change their shape
along the submanifold, which is the typical situation in the applications to
quantum wave guides and to quantum molecular dynamics.
  In order to explain the meaning and the relevance of some of the terms in the
effective Hamiltonian, we analyze in some detail the application to quantum
wave guides, where $\mathcal{C}$ is a curve in $\mathcal{A}=\mathbb{R}^3$. This
allows us to generalize two recent results on spectra of such wave guides.
