Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Tentative Contents of the Planned Sequel (Kuchment, 2027)
- 1 Periodic Ordinary Differential Operators
- 2 Multidimensional Periodicity: Lattices, Fundamental Domains, and Fourier Series
- 3 Floquet Transform and Direct Integral Decomposition
- 4 Dispersion Relations: Bloch, Fermi, and Floquet Varieties
- 5 Spectral Structure of Periodic Elliptic Operators
- 6 Localized Perturbations of Periodic Operators
- 7 Wannier Functions
- 8 Operators on Abelian Coverings of Compact Manifolds
- Appendix A Some Information from Complex Analysis
- Appendix B Some Information from Functional Analysis and Operator Theory
- Appendix C Operator Functions
- Appendix D Banach (Locally Trivial) Vector Bundles
- Appendix E The Landis Conjecture
- Appendix F Proofs of Some Technical Statements
- References
- Index
3 - Floquet Transform and Direct Integral Decomposition
Published online by Cambridge University Press: 31 July 2026
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Acknowledgements
- Introduction
- Tentative Contents of the Planned Sequel (Kuchment, 2027)
- 1 Periodic Ordinary Differential Operators
- 2 Multidimensional Periodicity: Lattices, Fundamental Domains, and Fourier Series
- 3 Floquet Transform and Direct Integral Decomposition
- 4 Dispersion Relations: Bloch, Fermi, and Floquet Varieties
- 5 Spectral Structure of Periodic Elliptic Operators
- 6 Localized Perturbations of Periodic Operators
- 7 Wannier Functions
- 8 Operators on Abelian Coverings of Compact Manifolds
- Appendix A Some Information from Complex Analysis
- Appendix B Some Information from Functional Analysis and Operator Theory
- Appendix C Operator Functions
- Appendix D Banach (Locally Trivial) Vector Bundles
- Appendix E The Landis Conjecture
- Appendix F Proofs of Some Technical Statements
- References
- Index
Summary
In Chapter 3, the main notions and techniques of multi-dimensional Floquet theory are introduced: Bloch and Floquet functions and the Floquet transform. The latter is the analog for the periodic case of the Fourier transform. Direct integral decomposition is introduced and studied. Important for the whole theory, the analogs of Plancherel and Paley–Wiener theorems are established. Analogously with the constant-coefficient case, these are invaluable tools throughout the book.
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- Periodic Elliptic Partial Differential OperatorsThe Main Structures, pp. 36 - 67Publisher: Cambridge University PressPrint publication year: 2026
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