Level statistics for one-dimensional Schrödinger operators and Gaussian beta ensemble

Abstract

We study the level statistics for two classes of 1-dimensional random Schrödinger operators : (1) for operators whose coupling constants decay as the system size becomes large, and (2) for operators with critically decaying random potential. As a byproduct of (2) with our previous result \cite{KN} imply the coincidence of the limits of circular and Gaussian beta ensembles.

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• The Sine${}_{\beta }$ operator
• Logarithmic, Coulomb and Riesz energy of point processes
• Eigenvectors of the critical 1-dimensional random Schroedinger operator
• Operator limit of the circular beta ensemble
• Fluctuation of density of states for 1d Schrödinger operators
• Uniform point variance bounds in classical beta ensembles
• Shape of eigenvectors for the decaying potential model
• Limiting distribution of extremal eigenvalues of d-dimensional random Schrödinger operator
• Curvature bound of Dyson Brownian Motion
• More scaling limits for 1d random Schrödinger operators with critically decaying and vanishing potentials