  An important result by Agmon implies that an eigenfunction of a
Schr\"{o}dinger operator in $\mathbb{R}^n$ with eigenvalue $E$ below the bottom
of the essential spectrum decays exponentially if the associated classically
allowed region $\{x \in \mathbb{R}^n~:~ V(x) \leq E \}$ is compact. We extend
this result to a class of Schr\"{o}dinger operators with eigenvalues, for which
the classically allowed region is not necessarily compactly supported: We show
that integrability of the characteristic function of the classically allowed
region with respect to an increasing weight function of bounded logarithmic
derivative leads to $L^2$-decay of the eigenfunction with respect to the same
weight. Here, the decay is measured in the Agmon metric, which takes into
account anisotropies of the potential. In particular, for a power law (or,
respectively, exponential) weight, our main result implies that power law (or,
respectively, exponential) decay of "the size of the classically allowed
region" allows to conclude power law (or, respectively, exponential) decay, in
the Agmon metric, of the eigenfunction.
