Geodesic Beams in Eigenfunction Analysis

This book discusses the modern theory of Laplace eigenfunctions through the lens of a new tool called geodesic beams. The authors provide a brief introduction to the theory of Laplace eigenfunctions followed by an accessible treatment of geodesic beans and their applications to sup norm estimates, L^p estimates,  averages, and Weyl laws.  Geodesic beams have proven to be a valuable tool in the study of Laplace eigenfunctions, but their treatment is currently spread through a variety of rather technical papers. The authors present a treatment of these tools that is accessible to a wider audience of mathematicians. Readers will gain an introduction to geodesic beams and the modern theory of Laplace eigenfunctions, which will enable them to understand the cutting edge aspects of this theory. The authors developed a framework in which an eigenfunction is decomposed as a sum of what are called geodesic beams near the point x. In broad terms, a geodesic beam is a piece of an eigenfunction that has been localized to a segment of geodesic that runs through x. This localization is accomplished using semiclassical analysis. Remarkably, this framework allows for the treatment of several problems related to eigenfunction concentration, including estimates for their norms, averages over submanifolds, and remainders in both pointwise and integrated Weyl laws.