Shape of Eigenvectors for the Decaying Potential Model

Abstract

We consider the 1d Schrödinger operator with decaying random potential and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions, which is based on the formulation by Rifkind and Virág (Geom Funct Anal 28:1394–1419, 2018). As a result, we have completely different behavior depending on the decaying rate $$\alpha > 0$$ of the potential: The limiting measure is equal to (1) Lebesgue measure for the supercritical case ( $$\alpha > 1/2$$ ), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case ( $$\alpha =1/2$$ ), and (3) the delta measure with its atom being uniformly distributed for the subcritical case ( $$\alpha <1/2$$ ). This result is consistent with the previous study on spectral and statistical properties.