Concentration of eigenfunctions of Schroedinger operators

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 $(1-\mathrm{\mid }x{\mathrm{\mid }}^{\beta }{)}^{-1\mathrm{/}2}$ when the non-perturbed potential resembles $\mathrm{\mid }x{\mathrm{\mid }}^{\beta }$, $\beta >0$, 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.