  We consider the limit measures induced by the rescaled eigenfunctions of
single-well Schr\"odinger operators. We show that the limit measure is
supported on $[-1,1]$ and with the density proportional to
$(1-|x|^\beta)^{-1/2}$ when the non-perturbed potential resembles $|x|^\beta$,
$\beta >0$, for large $x$, and with the uniform density for super-polynomially
growing potentials. We compare these results to analogous results in orthogonal
polynomials and semiclassical defect measures.
