Introduction to Isospectrality

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 Riemannian Manifold
1.5.1 Notation and Terminology
1.5.2 Modes of Vibration
1.5.3 The Spectral Theorem
1.5.4 Theorem (Spectral Theorem)
1.6 Solutions of the Wave Equation and Natural Sounds
1.6.1 Solutions of theWave Equation.
1.6.2 Natural Sounds.
1.6.3 Definition.
1.6.4 Proposition.
1.6.5 Comment.
1.7 Comparing Natural Sounds
1.7.1 Definition.
1.7.2 Theorem.
1.7.3 Comments
1.7.4 Conclusion.
2 Isospectrality and Isometry
2.1 Direct and Inverse Problems in Spectral Geometry
2.2 Interpreting Kac’s Question
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 Specβ (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
5.2.1 Theorem (Bers).
5.2.2 Corollary (Cheng).
5.3 Nodal Sets.
5.3.1 Nodal Domains of an Eigenfunction
5.3.2 Theorem (R. Courant).
5.3.3 Corollary.
5.3.4 Example of Eigenfunctions for the First Two Eigenvalues
6 Construction of Isospectral Flat Surfaces
6.1 Sunada’s Method
6.1.1 The Trace Formula
6.1.2 Proposition (The Trace Formula).
6.1.3 Gassmann Triples and Sunada’s Theorem
6.1.4 Definition.
6.1.5 Theorem (Sunada [70], 1985)
6.1.6 Comments.
6.2 Transplantations and Enhancements of Sunada’s Method
6.2.1 Theorem.
6.2.2 Comments.
6.2.3 Buser and Bérard Enhancements of Sunada’s Method
6.3 Construction of Isospectral Surfaces
6.3.1 Examples of Gassmann Triples
6.4 Cayley Graphs and Tessellated Buser Surfaces
6.4.1 Schreier Graphs.
6.5 Buser Flat Surfaces
6.6 The Gordon–Webb–Wolpert Domains
6.7 Transplantations
6.8 Bérard–Buser’s Surfaces and Beyond
7 General References
Chapter 2 The Wave Equation on Flat Manifolds
1 Flat Riemannian Manifolds
1.1 Euclidean Atlases and Flat Manifolds
1.1.1 Terminology.
1.1.2 Definitions.
1.1.3 Exercise.
1.1.4 Theorem.
1.1.5 Definition.
1.2 The Path-Distance on a Flat Manifold
1.3 Functions on a Flat Manifold
1.3.1 Definition.
1.3.2 Proposition.
2 The Wave Equation on a Flat Manifold
2.1 The Laplacian on a Flat Manifold
2.1.1 Theorem.
2.2 The Wave Equation and the Spectrum of a Flat Manifold
2.2.1 Natural Sounds of a Flat Manifold
2.2.2 Comment.
2.2.3 Definition.
2.2.4 Theorem.
3 Flat Surfaces with Piecewise Linear Boundary
3.1 Open Euclidean Sets with PL-Boundary
3.1.1 PLB-Domains of R2.
3.1.2 Differentiability on PLB-Domains of R2.
3.1.3 Proposition.
3.1.4 The Normal Derivative on the Boundary
3.2 Differentiability on PLB-Domains of R2.
3.2.1 Definition.
3.2.2 Theorem and Definition.
3.2.3 Open PLB-domains.
3.3 PLB-Surfaces
3.3.1 Definition
3.3.2 The Interior Gluing Data ε
3.4 PLB-Surface Defined by Gluing Open PLB-Domains
3.4.1 Definition.
3.4.2 Theorem.
3.4.3 Complete PLB-Atlas of
3.4.4 Topological Separation of M(ε)
3.4.5 Proposition.
3.4.6 Example.
3.4.7 Theorem and Definitions.
3.4.8 Comment.
3.4.9 Examples of Constructions of PLB-Surfaces
3.5 Differentiable Functions and the Laplacian on a PLB-Surface
3.5.1 Definitions.
3.5.2 Proposition and Definitions
3.5.3 Theorem.
4 Group Quotients of PLB-Surfaces
4.1 Theorem.
5 The Wave Equation of a PLB-Surface
5.1 The Extended Spectrum of a PLB-Surface
5.2 Isospectrality of PLB-Surfaces
5.2.1 Definition.
5.2.2 Theorem
6 Orbifold and Folding Boundaries
6.1 On the Action of a General Isometry of a PLB-Surface
6.1.1 Proposition.
6.1.2 Comment.
6.1.3 Corollary.
6.1.4 Comment.
Chapter 3 The Sunada–Bérard–Buser Method
1 The Sunada–Bérard–Buser Theorem for PLB-Surfaces
1.1 A Warning on the Notation for Left and Right Actions
1.2 The Sunada–Bérard–Buser Method in Brief
1.2.1 Transplantation Theorem
1.2.2 The Sunada–Bérard–Buser Theorem.
2 PLB-Surfaces with Specified Group of Isometries
2.1 The Cayley Graph of a Group
2.1.1 Convention.
2.1.2 Definition.
2.1.3 Examples of Cayley Graphs of Groups
2.1.4 Lemma.
2.1.5 Definition
2.1.6 Proposition.
2.1.7 Exercise.
2.2 The Schreier Graph of a Right G-Set
2.2.1 The Category of Right G-Sets
2.2.2 Definitions.
2.2.3 Proposition.
2.2.4 The Schreier Graph Functor on the Category Set-G
2.2.5 Definitions.
2.2.6 Theorem.
2.2.7 Examples of Schreier Graphs of Irreducible Right G-Sets
The Schreier graph (R\D4,S)
2.2.8 Comments.
2.3 Gassmann Triples
2.3.1 Proposition.
2.3.2 Exercises.
2.3.3 The Perlis–Brooks–Tse Gassmann Triple (G,Γ1,Γ2)
2.3.4 Proposition.
2.3.5 Comment.
2.3.6 The Schreier Graphs Associated with (G,Γ1,Γ2)
2.4 Piecewise Strings Associated with Schreier Graphs
2.4.1 One-Dimensional Tiles
2.4.2 The Buser Space M(1,X,S)
2.4.3 Remark.
2.4.4 Definition.
2.3.3 Definition.
2.4.5 Theorem.
2.5 PLB-Surfaces Associated with Schreier Graphs
2.5.1 Two-Dimensional Tiles
2.5.2 The Buser Surfaces
2.5.3 Comments.
2.5.4 Examples
2.5.5 Theorem.
2.6 Buser Manifolds AssociatedWith Schreier Graphs
2.7 Volume and Perimeter of
2.7.1 Proposition.
2.8 On the Buser Method
2.8.1 Irregular Tiles.
2.8.2 Proposition.
2.8.3 Comment.
3 Non-Isometric Isospectral Buser Spaces
3.1 Theorem.
3.1.1 Comment.
3.2 Buser Isospectral Spaces
4 Transplantations in Buser Surfaces
4.1 Transplantations and Γ2 X Γ1-Orbits
4.1.1 Finding Transplantations for General Triples (G,Γ1,Γ2)
4.2 Transplantation Matrices on Buser’s Isospectral Spaces
4.2.1 Proposition.
4.3 The Name of the Game
4.3.1 Manual Check of the Differentiability of Transplantations
4.3.2 Comment.
4.3.3 Exercise.
4.3.4 Exercise.
4.4 Reflections in Buser Surfaces
4.4.1 Definition.
4.4.2 Proposition.
4.4.3 Comment.
Chapter 4 The Gordon–Webb–Wolpert Isospectral Domains
1 A Special Reflection on Buser Surfaces
1.1 Notation.
1.2 The Reflections σi.
1.3 The GWW Domains Di
1.3.1 Folding Edges.
1.3.2 Comment.
1.3.3 Proposition (GWW[39])
2 Transplantations of Continuous Functions
2.1 Proposition.
2.2 Double-Sided Tessellation of Buser Surfaces
2.3 The Transplantation Matrix for Continuous Functions on GWWDomains
3 Lifting Smooth Functions of GWW Domains
3.1 The Euclidean Half-Space
3.1.1 Theorem.
3.1.2 Comment.
3.2 The Spectrum of D := (σ)\M, for M := R2 and 2(x,y) := (−x,y)
3.2.1 Corollary.
3.3 Different Natures of the Boundaries on GWW Domains
3.4 Mixed Boundary Conditions on GWWDomains
3.4.1 Definitions.
3.4.2 Theorem
4 Isospectrality of Gordon–Webb–Wolpert Domains
4.1 The Mixed Extended Spectrum of a GWWDomain
4.1.1 Theorem (GWW[39]).
5 Transplantations in Gordon–Webb–Wolpert Domains
5.1 General Transplantation Setup
5.2 Transplantation Matrix TN
5.3 Transplantation Matrix TD
5.4 Transplantation Recipe
5.5 Changing the Tile’s Shape
5.5.1 The Hen and the Arrow.
5.5.2 Exercises.
6 Research in Isospectrality
6.1 Important Dates in Isospectrality.
6.2 Other Routes of Research
6.2.1 On Transplantations
6.2.2 On the Arithmetical Approach to Isospectrality
6.2.3 On Isospectral Simply-Connected Closed Manifolds
6.2.4 On Isospectral Compact Flat Manifolds
6.3 On Non-Strongly Isospectral Manifolds
6.3.1 Miscellanea
Appendix A Linear Representations of Finite Groups and Almost-Conjugate Subgroups
1 The Category of Linear Representations of a Group
1.1 The Group Ring.
1.1.1 Proposition.
1.1.2 Definition.
1.2 Category of M-Modules
1.2.1 Definitions.
1.2.2 Exchange Principle
1.3 Permutation Representations
1.3.1 Vector Space Spanned by a Set
1.3.2 M-Module Spanned by a M-Set
1.3.3 Examples
1.4 Symmetrization Operator
1.4.1 Definition.
1.4.2 Proposition
1.5 Operations on M-Modules
1.5.1 Products and Direct Sums
1.5.2 Submodules and Quotient Modules
1.5.3 Proposition.
1.5.4 The M-Module HomQ(_,_)
1.5.5 Proposition.
1.5.6 The Module (_)_ := HomQ(_,Q)
1.5.7 The Functor HomM(Q[G],_)
1.5.8 Proposition.
1.6 Semisimplicity of Finite-Dimensional Representations
1.6.1 Definitions
1.6.2 Theorem.
1.6.3 Exercise.
1.7 Unitary M-Modules
1.7.1 Definitions.
1.7.2 Proposition.
1.7.3 Unitary Equivalences
1.7.4 Proposition.
1.7.5 Unitarity of Permutation Representations
1.7.6 Exercise.
1.7.7 Proposition.
1.8 Characters of Finite-Dimensional M-Modules
1.8.1 Definitions
1.8.2 Proposition
1.9 Characters and Dimensions of Invariant Subspaces
1.9.1 Theorem.
1.10 Schur Orthogonality Relations
1.10.1 Theorem
1.10.2 Comments.
1.11 Characters of Permutation Representations
1.11.1 Proposition.
1.11.2 Exercise.
2 Conjugate and Almost-Conjugate Subgroups
2.1 Almost-Conjugate Subgroups
2.1.1 Definition.
2.1.2 Theorem.
2.2 Conjugate Subgroups
2.3 Almost-Conjugate Subgroups and Invariants
2.3.1 Theorem.
3 Transplantations
3.1 Making Morphisms Explicit
3.1.1 Proposition.
3.1.2 Transplantation Theorem.
3.1.3 Exercises.
3.2 Transplantations and Double-Cosets
3.2.1 Lemma.
Appendix B The Laplacian as Isometry-Invariant Differential Operator
1 Affinity of Euclidean Isometries
1.1 The Euclidean Structure of Rn
1.1.1 Euclidean Linear Maps and the Orthogonal Group
1.1.2 Terminology.
1.1.3 Exercises.
1.2 Linearity of Euclidean Isometries Fixing the Origin
1.2.1 Global and Local Isometries
1.2.2 Proposition.
1.3 Affinity of Euclidean Isometries
1.3.1 Definitions.
1.3.2 Proposition and definitions.
1.3.3 Theorem.
1.3.4 The Decomposition of the Euclidean Group
1.3.5 Theorem and Definitions.
2 Differentiable Maps on Open Euclidean Sets
2.1 Algebra of Real Functions on a Set
2.2 Algebra of Continuous Functions
2.2.1 Proposition.
2.3 Algebra of Differentiable Functions on Open Euclidean Sets
2.3.1 Directional Derivatives on Open Euclidean Sets
2.3.2 Definitions.
2.3.3 Proposition.
2.4 Differentiable Maps Between Open Euclidean SetsFunctions.
2.4.1 Coordinate Functions.
2.4.2 Definitions.
2.4.3 Tangent Maps
2.4.4 Proposition.
2.4.5 Comment.
2.4.6 Corollary.
2.4.7 Schwarz Theorem.
2.4.8 The Pullback
2.4.9 Proposition
2.4.10 Comment.
2.4.11 Proposition.
3 Linear Differential Operators on Open Euclidean Sets
3.1 The Noncommutative Algebra EndR (C∞(U)
3.1.1 Some Basic Operators
3.1.2 Notational Convention.
3.1.3 Definitions.
3.1.4 Proposition.
3.1.5 Restriction of Linear Differential Operators
3.1.6 Theorem.
3.2 Transfer of Differential Operators
3.2.1 Definition.
3.2.2 Proposition and Definition.
3.2.3 Comment.
3.2.4 Theorem and Definition.
3.2.5 Transfer Maps Associated with Locally Affine Isomorphisms
3.2.6 Proposition.
3.2.7 Exercise.
4 Invariant Differential Operators
4.1 Group of Diffeomorphisms of an Open Euclidean Set
4.1.1 Definitions.
4.1.2 Exercises.
4.1.3 Isometry-Invariance of the Laplacian
4.1.4 Theorem.
4.1.5 Comments.
Appendix C The Path-Distance on a Hausdorff Connected Flat Manifold
1 The Length of a Path on a Flat Manifold
1.1 Terminology and Notation.
1.1.1 Proposition.
1.1.2 Definition.
1.1.3 Exercises.
1.2 The Path-Distance on a Connected Flat Manifold
1.2.1 Theorem.
1.2.2 Exercises.
1.2.3 Definition.
1.2.4 Comments.
2 The Length of a Differentiable Path on a Flat Manifold
2.1 The Length of a Differentiable Path on Rd
2.2 The Norm of the Derivative of a Differential Path
2.2.1 Proposition.
2.2.2 Comment.
3 Local Riemannian Isomorphisms of Flat Manifolds
3.1 Orthogonality of Tangent Maps
3.1.1 Definition.
3.1.2 Theorem.
3.1.3 Comment.
Appendix D Group Quotients of Flat Manifolds
1 Quotients of Topological Spaces
1.1 The Universal Property of the Quotient Topology
1.1.1 Definition.
1.1.2 Theorem
2 Group Quotients of Flat Manifolds
2.1 Coverings, Group Actions and Group Quotients
2.1.1 Definitions and Terminology.
2.1.2 Proposition.
2.1.3 Theorem.
2.1.4 Exercise.
2.2 Factorization of Group Quotients
2.2.1 Proposition.
2.2.2 Exercise.
3 Examples of Flat Surfaces
3.1 (A) Quotients of R2 by of Four Groups of Isometries
3.1.1 N-Orbits.
3.1.2 Proposition.
3.1.3 Fundamental Domains and Tessellations
3.1.4 Definition.
3.1.5 Exercise.
3.1.6 Proposition.
3.1.7 Exercise.
3.1.8 Compact Fundamental Domains
3.1.9 Noncompact Fundamental Domains
3.1.10 Exercise.
3.1.11 Open Tubular Neighborhoods of Fundamental Domains
3.1.12 Proposition.
3.2 (B) Gluing Euclidean Atlases
3.2.1 Comments.
References
Glossary
Index