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 [2] imply the coincidence of the limits of circular and Gaussian beta ensembles.

The maximum deviation of the $\text{Sine} _\beta $ counting process

“…It should be noted there is another SDE description due to [KS09] (only recently proven to give rise to the same process by [Nak14], while another proof follows from [VV17b]), which can be related to (1) by a time-reversal. This arises due to an order reversal of the Prüfer phases, for which reason the correlation structure is reversed from the previously studied CβE model.…”

Poisson statistics for 1d Schrödinger operators with random decaying potentials

“…Remark 1.1 When we consider two reference energies E 1 , E 2 , E 1 = E 2 , then the corresponding point processes ξ 1 , ξ 2 jointly converge to the independent Poisson processes of intensity dλ/π. Remark 1.2 Together with results in [7,11], we have 2…”

“…The 1d Schrödinger operators with random decaying potentials are known to have rich spectral properties depending on the decay order of the potentials (e.g., [8,6]). Recently, the level statistics problem of this operators are studied and turned out to be related to the β-ensembles which appear in the random matrix theory [5,9,7,11]. In this paper we consider the following Hamiltonian.…”

The Sine $$_\beta $$ β operator

“…In Section 8 we use the connection between hyperbolic carousels and differential operators to show that the point process scaling limit of the circular β-ensembles (the CβE process) is the same as the Sine β process. Nakano (2014) has recently proved this equivalence by deriving both processes as the limit of the same sequence of models.…”