Geodesic Beams in Eigenfunction Analysis

ISBN-13: 9783031315855 / Angielski

Yaiza Canzani; Jeffrey Galkowski

Opis

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.

Spis treści

Introduction.- The Laplace operator.- Axiomatic introduction to semiclassical analysis.- Basic properties of eigenfunctions and eigenvalues.- The Koch–Tataru–Zworski approach to L∞ estimates.- Geodesic Beam Tools.- Applications of the geodesic beam decomposition.- Dynamical ideas.

O autorze

Yaiza Canzani, Ph.D., is Associate Professor in the Department of Mathematics at the University of North Carolina at Chapel Hill. She received her Ph.D. in Mathematics at McGill University. After graduating, Dr. Canzani held postdoctoral positions at Harvard University and the Institute for Advanced Study. Her work has been recognized with a Sloan Fellowship, an NSF CAREER grant, and the AWM Sadosky Prize in Analysis.

Jeffrey Galkowski, Ph.D., is Professor in the Department of Mathematics at University College London. He received his Ph.D. in Mathematics at University of California at Berkeley. Dr. Galkowski held a NSF Postdoctoral Fellowship at Stanford University and the CRM-ISM postdoctoral fellowship at McGill University. His work has been recognized with an EPSRC early career fellowship as well as the Adams Prize in Mathematics from the University of Cambridge.

Blurb

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 beams 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.

This book:

• Reviews several physical phenomena related to Laplace eigenfunctions, ranging from the propagation of waves to the location of quantum particles;
• Introduces the cutting edge theory and microlocal methods of geodesic beams;
• Discusses how eigenfunctions of the Laplacian play a crucial role both in physics and mathematics.