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Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M , g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(-\Delta _g)$$ 1 [ 0 , λ 2 ] ( - Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{-n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n-1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π ) - n vol R n ( B ) vol g ( W ) λ n + O ( λ n - 1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the off-diagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under non-looping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.
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Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle
We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $$2\lambda \cos\2π(j^2\omega+jy+x)$$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $$\lambda>0$$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $$L(\lambda)$$ at small $$\lambda$$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $$L(\lambda)$$ is fully consistent with $$L(\lambda)$$ being positive and satisfying the usual Figotin-Pastur type asymptotics $$L(\lambda)\sim C\lambda^2$$ as $$\lambda\to 0$$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $$\lambda<1$$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.
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- Award ID(s):
- 1800689
- PAR ID:
- 10105665
- Date Published:
- Journal Name:
- Journal of spectral theory
- ISSN:
- 1664-039X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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