# Eigenfunction concentration via geodesic beams

## TL;DR

This paper introduces a new technique for analyzing the concentration of Laplace eigenfunctions by decomposing them into geodesic beams, leading to improved bounds on their norms and pointwise behavior.

## Contribution

The paper develops a novel method using geodesic beam decomposition to study eigenfunction concentration, providing quantitative improvements over existing bounds.

## Key findings

- Improved bounds on $L^
Infty$ norms of eigenfunctions.
- Enhanced estimates for eigenfunction averages over submanifolds.
- Quantitative bounds depending on the dynamical property T.

## Abstract

In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.