Geodesic Beams in Eigenfunction Analysis
Geodesic Beams in Eigenfunction Analysis
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特征函数分析中的测地梁
DOI：
10.1007/978-3-031-31586-2
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
Canzani Y
中科院分区：
文献类型：
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作者：
Canzani Y
这本书的目的是解释概念背后的测地线梁方法，我们已经开发研究的行为，高能量本征函数。测地线梁的想法受到Koch-Tataru-Zworski [KTZ 07]以及第二作者Toth [GT 17]的工作的启发。测地线束最初出现在[Gal 19]中，并在[CG 19 b，CG 21，CG 19 a，CG 20 b]中发展成目前的形式。这本书的目的是要访问研究生，并针对读者有兴趣研究的行为特征函数。虽然熟悉半经典分析将有助于阅读这篇文章，我们提供了一个快速，公理化介绍的主题在第一章。3.这一章，原则上，包含了所有必要的半经典分析结果，以了解其余的文本。关于半经典机械的更详细的论述，我们请读者参阅书籍[Zwo 12，DS 99，DZ 19]。在整本书中，我们选择在充分证明能够说明重要思想并对我们的方法至关重要的时候给出充分证明。相反，当技术细节可能会掩盖想法时，我们会提出证明的大纲。在这种情况下，我们建议读者参考相应的文章，以获得证明的全部细节。我们希望我们的选择将使文本相对容易阅读，同时仍然包含我们分析的要点。我们介绍的各个部分，这段文字在很大程度上影响了Zworski的书[Zwo 12]的作者首先学会了半经典分析。我们的半经典分析方法是通过与Maciej Zworski，Semyon Dyatlov和Andras Vasy的长期互动而获得的，我们对本征函数分析的观点是由Dima Jakobson，Peter Sarnak，John Toth和Steve Zelditch指导的。没有他们的支持，这一案文是不可能的。Y. Canzani得到了Alfred P. Sloan基金会，NSF CAREER Grant DMS-2045494和NSF Grant DMS-1900519的支持。J. Galkowski感谢EPSRC通过赠款EP/V001760/1和EP/V051636/1提供的支持。
This book aims to explain the concepts behind the geodesic beam method that we have developed to study the behavior of high energy eigenfunctions. The idea for geodesic beams was inspired by the work of Koch-Tataru-Zworski [KTZ07] as well as the second author’s work with Toth [GT17]. Geodesic beams originally appeared in [Gal19] and were developed into their current form in [CG19b, CG21, CG19a, CG20b]. The book is intended to be accessible to graduate students and is aimed at readers interested in studying the behavior of eigenfunctions. Although familiarity with semiclassical analysis will be helpful when reading this text, we provide a quick, axiomatic introduction to the subject in Chap. 3. This chapter, in principle, contains all of the semiclassical analysis results necessary to understand the rest of the text. For more detailed treatments of the semiclassical machinery, we refer the reader to the books [Zwo12, DS99, DZ19]. Throughout the book, we chose to present full proofs when they illustrate important ideas and are central to our method. We instead present outlines of proofs when the technical details can obscure the ideas. In this case, we refer the reader to the corresponding article for the full details of the proof. We hope that our choices will make the text relatively easy to read, while still containing the main points of our analysis. Our presentation of various parts of this text was heavily influenced by Zworski’s book [Zwo12] from which the authors first learned semiclassical analysis. Our approach to semiclassical analysis was informed by long interactions with Maciej Zworski, Semyon Dyatlov, and Andras Vasy, and our perspective on the analysis of eigenfunctions has been guided by Dima Jakobson, Peter Sarnak, John Toth, and Steve Zelditch. This text would not have been possible without their support. Y. Canzani was supported by the Alfred P. Sloan Foundation, NSF CAREER Grant DMS-2045494, and NSF Grant DMS-1900519. J. Galkowski acknowledges support from the EPSRC through grants EP/V001760/1 and EP/V051636/1.