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Topics in Material Science: Spectral theory in wave dynamics, from classical theory to open problems
Louisiana State University
Spring Semester, 2017
Prof. Stephen Shipman
Place: Lockett 111
Time: 1:30-2:50 Tuesday and Thursday
Office: Room 314 of Lockett Hall
Office Hours: By appointment
Spectral theory of self-adjoint and normal operators underlies the dynamics of linear waves in classical and quantum physics. In fact, the mathematical theory is largely driven by physics. This is a huge and varied field, and I will choose the direction, flavor, and topics based on my research activity and tastes.
The material will come from my notes and from various books and articles in the literature.
The prerequisites are real and complex analysis. Spectral theory and some PDEs will also be useful but will be introduced as needed.
Proposed Course Topics
- Spectral theorem and concrete realizations of it
- Unitary representations and Fourier transforms for symmetry groups of a physical structure
- Connection to complex analysis and moment problems
- Spectral theory in wave dynamics
- Operators for wave dynamics that commute with symmetry groups
- Analysis of operator resolvents
- Scattering of waves by objects: modified Fourier modes caused by local perturbations
- Limiting amplitude principle and limiting absorption principle
- Some relatively classical examples
- Bound states and eigenvalues, symmetry-protected spectrally embedded eigenvalues
- Coercivity and compactness
- The Fermi algebraic surface for periodic media, reducibility, relation to embedded eigenvalues
- Current research and open problems
- "Negative-index" materials
- Non-symmetry-protected embedded eigenvalues
- Topologically protected eigenstates
- Symmetries of Euclidean space--the wallpaper groups and the Bieberbach groups
- Nonlinear scattering problems
We will not use any one source as a text book. My lectures will draw from my notes and several references.
There is a bibliography of relevant works below, with links to some PDF files of excerpts.
Enrolled students will present a special topic or problem to the class. There will be lots of problems to solve, of course, but no required submission of assignments.
Evaluation of performance in the course is based on the presentation.
Lecture Notes and Problem Sets
Notes 1: Introduction
Notes 2: Motivation: The spectral theorem in finite dimension
Notes 3: Integral representations of analytic functions
Notes 4: Unbounded self-adjoint operators in Hilbert space
Notes 5: Spectral theorem for the resolvent of a self-adjoint operator
Notes 6: Spectral theory and Fourier analysis
Notes 7: Symmetry, Fourier analysis, and spectral theory
Notes 8: The Floquet transform for discrete translation Groups
Notes 9: Periodic operators
Notes 10: Local perturbation and modified Fourier transform
Appendix: Riemann-Stieljes integral and Helly theorems
Peter Kuchment, An Overview of Periodic Elliptic Operators,
Bulletin of the AMS, 2016.
My Notes on Resonance 2014.
Peter Kuchment, The Mathematics of Photonic Crystals, Chapter 7 in Mathematical Modeling in Optical Science, Vol. 22 of Frontiers in Applied Mathematics, SIAM 2001.
David Bindel, Resonance Sensitivity for Schrödinger, Notes, 2006.
David Bindel and Maciej Zworski, Theory and Computation of Resonances in 1D Scattering (website).
Maciej Zworski, Resonances in Physics and Geometry, Notices of the AMS, 1999.
Maciej Zworski, Mathematical Theory of Scattering Resonances Lecture notes, Version 0.1, 2016.
Maciej Zworski Mathematical Study of Scattering Resonances 2016.
Michael Reed and Barry Simon,
Methods of Modern Mathematics:
Vol. I Functional Analysis,
Vol. II Fourier Analysis and Self-Adjointness,
Vol. III Scattering Theory,
Vol. IV Analysis of Operators,
excerpts from Vol. I (spectral theorem),
Vol. II (self-adjoint extensions),
Academic Press, 1980.
Stephen P. Shipman and Aaron T. Welters,
Resonant electromagnetic scattering in anisotropic layered media,
J. Math. Phys. 54(10) 103511-1--40 (2013).
N. I. Akhiezer and I. M. Glazman,
Theory of Linear Operators in Hilbert Space,
Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.
A. N. Kolmogorov and S. V. Fomin,
Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).
Gerald B. Folland,
Introduction to Partial Differential Equations,
Second Edition, Princeton University Press, 1995.
Excerpt: Chapter 0.