Fluctuation of density of states for 1d Schr\"odinger operators

TL;DR

This paper investigates how the density of states for one-dimensional Schrödinger operators with random decaying potentials varies asymptotically, revealing distinct behaviors depending on the decay rate parameter.

Contribution

It provides the second term asymptotics of the density of states for 1d Schrödinger operators with random decaying potentials, highlighting differences across decay regimes.

Findings

Different asymptotic behaviors for $ ho( heta)$ depending on $eta$

Substantial differences in density of states for $eta > 1/2$, $eta < 1/2$, and $eta = 1/2$

New insights into spectral properties of 1d Schrödinger operators with decaying randomness

Abstract

We consider the 1d Schr\"odinger operator with random decaying potential and compute the 2nd term asymptotics of the density of states, which shows substantial differences between the cases $\alpha > \frac 12$, $\alpha < \frac 12$ and $\alpha = \frac 12$.