Scaling asymptotics for ladder sequences of spherical harmonics at caustic latitudes

TL;DR

This paper investigates the asymptotic behavior of ladder sequences of spherical harmonics at caustic latitudes, revealing Airy scaling and describing their measure limits, advancing understanding of their concentration phenomena.

Contribution

It establishes Airy scaling asymptotics for ladder sequences and characterizes the weak* limits of their empirical measures on caustic latitude circles.

Findings

Ladder sequences exhibit Airy scaling asymptotics at caustic latitudes.

The weak* limit of empirical measures of spherical harmonics restrictions is characterized.

Provides new insights into the concentration of spherical harmonics on specific geometric sets.

Abstract

We study the concentration of ladder sequences of spherical harmonics on caustic latitude circles. We prove that they have Airy scaling asymptotics. We also determine the weak* limit of certain empirical measures of $L^2$ norms of restrictions of spherical harmonics to these latitude circles.