Introduction to isospectrality
"Can one hear the shape of a drum?" This striking question, made famous by Mark Kac, conceals a precise mathematical problem, whose study led to sophisticated mathematics. This textbook presents the theory underlying the problem, for the first time in a form accessible to students. Specifically, this book provides a detailed presentation of Sunada's method and the construction of non-isometric yet isospectral drum membranes, as first discovered by GordonWebbWolpert. The book begins with an intr…
Read More ↓Title Statements
- Full title :Introduction to isospectrality / Alberto Arabia.
Summary
"Can one hear the shape of a drum?" This striking question, made famous by Mark Kac, conceals a precise mathematical problem, whose study led to sophisticated mathematics. This textbook presents the theory underlying the problem, for the first time in a form accessible to students. Specifically, this book provides a detailed presentation of Sunada's method and the construction of non-isometric yet isospectral drum membranes, as first discovered by GordonWebbWolpert. The book begins with an introductory chapter on Spectral Geometry, emphasizing isospectrality and providing a panoramic view (without proofs) of the SunadaBerardBuser strategy. The rest of the book consists of three chapters. Chapter 2 gives an elementary treatment of flat surfaces and describes Buser's combinatorial method to construct a flat surface with a given group of isometries (a Buser surface). Chapter 3 proves the main isospectrality theorems and describes the transplantation technique on Buser surfaces. Chapter 4 builds GordonWebbWolpert domains from Buser surfaces and establishes their isospectrality. Richly illustrated and supported by four substantial appendices, this book is suitable for lecture courses to students having completed introductory graduate courses in algebra, analysis, differential geometry and topology. It also offers researchers an elegant, self-contained reference on the topic of isospectrality.
Read More ↓Edition Statements
- Publication Date:2022.
Publication Statements
- Publisher(s):Springer
- Place of Publication:Cham
- Language:eng
- Publication Date:2022.
Notes
1.2 The Path-Distance on a Flat Manifold
The Physical Object
- Description:1 online resource (247 p.).
Edition Identifiers
- ISBN:9783031171239, 3031171233
- DOI:10.1007/978-3-031-17123-9
- oclc:on1344540094
- LC Classification:QA614.95
- Dewey Decimal:516.3/62
Table of Contents
2.3 Boundary Conditions and Irreducibility -- 3 The Spectra of Strings -- 3.1 Strings -- 3.2 Smooth Functions on Strings -- 3.3 The Extended Spectra of Strings -- 3.3.1 Proposition. -- 3.3.2 Exercises. -- 4 The Spectra of Rectangular Membranes -- 4.1 Domains and Membranes -- 4.2 Smooth Functions on Membranes -- 4.3 Rectangular Membranes -- 4.4 Dirichlet and Neumann Spectra of Rectangular Membranes -- 4.5 Notable Properties of Spec[beta] (D) -- 4.5.1 Proposition. -- 5 Some General Results of Spectral Geometry -- 5.1 Weyl's Law. -- 5.1.1 Exercise. -- 5.2 Local/Global Uniqueness of Eigenfunction
Intro -- Preface -- The Plan of the Book -- Acknowledgements -- Contents -- Chapter 1 Introduction -- 1 Spectral Geometry -- 1.1 Visible Light Spectroscopy -- 1.2 From Light to Sound -- 1.2.1 The Fourier Decomposition of Periodic Functions -- 1.2.2 Comment. -- 1.2.3 Fourier Transform of Functions -- 1.2.4 Comment. -- 1.3 The Wave Equation -- 1.3.1 Strings -- 1.3.2 Membranes and Manifolds -- 1.3.3 Manifolds with Boundary -- 1.3.4 On the Constant ' -- 1.4 Natural Vibrations and Natural Sounds -- 1.4.1 Natural Vibrations. -- 1.4.2 Proposition. -- 1.4.3 Natural Sounds.
1.5 The Spectrum of a Rie…
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