Level repulsion for Schrödinger operators with singular continuous spectrum

Abstract

We describe a family of half-line continuum Schrödinger operators with purely singular continuous spectrum on (0, ∞), exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum Christoffel-Darboux kernel to the sine kernel.

Smoothness of integrated density of states and level statistics of the Anderson model when single site distribution is convolution with the Cauchy distribution

“…For one-dimensional continuum decaying model, Nakano [5] showed that the statistics for α > 1 2 is clock and for α = 1 2 , it is circular β-ensemble, see also Kotani [38]. Avila-Last-Simon [34] showed quasi-clock behaviour for ergodic Jacobi operator in region of absolutely continuous spectrum and Breuer-Weissman [8] showed the strong level repulsion (uniform clock behaviour) for one dimensional continuum model with purely singular continuous spectrum.…”

On the spacing of zeros of paraorthogonal polynomials for singular measures

“…In light of the above results and discussion, it is natural to wonder whether singularity of µ implies less regularity of the asymptotic zero spacing. The example in [2] (see [3] for a continuum Schrödinger operator analog) shows that the situation in the case of R is more subtle. By considering the Jacobi coefficients associated with µ on R, [2] presents a family of purely singular measures where bulk universality, and therefore clock behavior, holds at every point of [−2, 2].…”

“…The observation is that the degree of continuity of the underlying measure, in terms of comparison with α-dimensional Hausdorff measure, implies a lower bound on the local spacing of the zeros. The second aim of this paper is to present the POPUC analog of an example on the real line [2,3] that shows that singular measures may still have strong asymptotic repulsion, implying that upper bounds coming from singularity of the measure are probably more subtle.…”