Log-scale equidistribution of zeros of quantum ergodic eigensections

Abstract

Under suitable hypotheses, a symplectic map can be quantized as a sequence of unitary operators acting on the $N$th powers of a positive line bundle over a Kähler manifold. We show that if the symplectic map has polynomial decay of correlations, then there exists a density one subsequence of eigensections whose masses and zeros become equidistributed in balls of logarithmically shrinking radii of lengths $|\log N{|}^{-\gamma }$ for some constant $\gamma >0$ independent of $N$.

arXiv commentThe ergodicity assumption is relaxed from exponential to polynomial decay of correlations

References in corpus (9)

• Distribution of zeros of random and quantum chaotic sections of positive line bundles
• Berezin-Toeplitz quantization on Kaehler manifolds
• Small scale quantum ergodicity in negatively curved manifolds
• Chaotic eigenfunctions in phase space
• Small scale equidistribution of eigenfunctions on the torus
• Upper bounds on the rate of quantum ergodicity
• On the asymptotic distribution of zeros of modular forms
• $L^p$ norms, nodal sets, and quantum ergodicity
• $C^r$-density of (non-uniform) hyperbolicity in partially hyperbolic symplectic diffeomorphisms