Log-scale equidistribution of zeros of quantum ergodic eigensections

TL;DR

This paper proves that for certain quantum systems with symplectic maps exhibiting polynomial decay of correlations, a large subset of eigensections have zeros and masses that become uniformly distributed in very small, logarithmically shrinking neighborhoods as the system size grows.

Contribution

It establishes the equidistribution of zeros of quantum eigensections at a logarithmic scale for symplectic maps with polynomial decay of correlations.

Findings

Existence of a density one subsequence of eigensections with equidistributed zeros.

Zeros become equidistributed in balls shrinking at a logarithmic rate.

Results apply to quantum systems modeled by symplectic maps with specific decay properties.

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 K\"{a}hler 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 $\lvert \log N \rvert^{-\gamma}$ for some constant $\gamma > 0$ independent of $N$.