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.
Author(s): Yaiza Canzani, Jeffrey Galkowski
Series: Synthesis Lectures on Mathematics & Statistics
Publisher: Springer
Year: 2023
Language: English
Pages: 122
City: Cham
Preface
Contents
1 Introduction
2 The Laplace Operator
2.1 Why Study Laplace Eigenfunctions and Eigenvalues?
2.1.1 Quantum Mechanics
2.1.2 Heat Propagation
2.1.3 Wave Propagation
2.1.4 Inverse Problems
2.2 Statement of the State of the Art Estimates
2.2.1 Linfty Norms
2.2.2 Averages
2.2.3 Weyl Laws
2.2.4 Lp Norms
3 Axiomatic Introduction to Semiclassical Analysis
3.1 Basic Definitions: Pseudodifferential Operators
3.1.1 Pseudodifferential Operators on mathbbRn
3.1.2 Pseudodifferential Operators on a Compact Manifold
3.1.3 Symbol Map
3.2 Wavefront Set and Microsupport
3.3 Ellipticity and Inverses
3.4 Egorov's Theorem
4 Basic Properties of Eigenfunctions and Eigenvalues
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4.1 The Semiclassical Laplacian
4.2 Smoothness and Existence of the Resolvent
4.3 Spectral Theorem
5 The Koch–Tataru–Zworski Approach to Linfty Estimates
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5.1 Basic Estimates
5.2 Factorization
5.3 The Hörmander Linfty Estimate
6 Geodesic Beam Tools
6.1 Basic Geodesic Beam Estimates
6.2 Good Covers and Partitions
6.2.1 Geodesic Tubes
6.2.2 Construction of Good Covers and Partitions
7 Applications of the Geodesic Beam Decomposition
7.1 Applications to Sup-Norms
7.1.1 The Linfty Estimate in Terms of Analytic Data
7.1.2 Error Term as a Function of the Laplacian
7.1.3 The Linfty Estimate in Terms of Dynamical Data
7.2 Applications to Averages
7.2.1 Remarks on the Analog of Theorem 7.2.2 for More General Operators
7.3 Applications to Norms
7.3.1 Discussion of the Proof of Theorem 7.3.1
7.4 Applications to Weyl Asymptotics
7.4.1 Outline of the Proof of Theorem 2.2.5摥映數爠eflinkt:laplaceWeyl2.2.52
7.4.2 The Tauberian Lemma
7.4.3 Technical Estimates on the Spectral Projector
7.4.4 Lipschitz Type Estimates for the Spectral Projector
7.4.5 The Fourier Representation to Compare Two Smoothed Counting Functions
7.4.6 Proof of Theorem 2.2.5摥映數爠eflinkt:laplaceWeyl2.2.52
7.4.7 Pointwise Weyl Asymptotics
8 Dynamical Ideas
8.1 Designing Coverings to Apply Volume Assumptions on Non-looping Directions
8.2 Transversality
8.2.1 Sketch of the Proof of Theorem 2.2.1摥映數爠eflinkt:noConj22.2.12
8.3 Contraction
References
