Concentration of Eigenfunctions of Schrödinger Operators

Abstract

AbstractWe consider the limit measures induced by the rescaled eigenfunctions of Schrödinger operators with even confining potentials. We show that the limit measure is supported on $$[-1,1]$$ [ - 1 , 1 ] and with the density proportional to $$(1-|x|^\beta )^{-1/2}$$ ( 1 - | x | β ) - 1 / 2 when the non-perturbed potential resembles $$|x|^\beta $$ | x | β , $$\beta >0$$ β > 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.