Topics in Material Science: Mathematics of Linear Wave Phenomena

SEC live-streamed graduate course

Math 7384-1
Louisiana State University and Auburn University
Fall Semester 2018 and Spring Semester 2019

Prof. Stephen Shipman (LSU) and Prof. Junshan Lin (Auburn)

Registration: Each student must register for the course at their home institution.

Live-streaming: Tuesday and Thursday 1:55-2:55 Central Time

The class will be live-streamed via Zoom. The software can be downloaded at this link. More information about the use of software can be found at this link.

A course website for participants at Auburn University is forthcoming.

For participants at LSU for Spring 2019:

Place: Lockett Hall 381
Time: 1:30-2:50 Tuesday and Thursday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@lsu.edu
Office Hours: By appointment

Course Description

This course will be taught jointly by Profs. Stephen Shipman (LSU) and Junshan Lin (Auburn University) in a live-streaming setting. Students from any SEC university may enroll. It is intended to be a two-semester course, although the first semester may be taken in isolation. The second semester run in the spring of 2019.

The material in this course builds toward open problems in mathematical physics centered around wave dynamics and scattering in electromagnetics and acoustics. The mathematical topics form a coherent body of theory and techniques. The main components are the partial differential equations (PDE) of electromagnetics and acoustics and other derived phenomena in complex media; integral equations and boundary-integral representations of solutions to PDE; Fourier analysis and the residue calculus for the study of scattering and resonance; spectral theory of differential and integral operators illustrated and motivated by examples; and asymptotic analysis. The second semester of the course will concentrate on specific problems that are motivated by modern scientific and mathematical research. A goal is that students will be able to understand open problems in this area and be equipped with the basic mathematical tools to begin trying to solve them.

Prerequisite

Course topics and notes for Spring 2019

Course topics for Fall 2018

1. Preliminary mathematical material
   
   
   • Measure and integration on the line: Increasing functions and the Helly Selection Theorem; functions of bounded variation; the Riemann-Stieljes integral and the Riesz Representation Theorem; signed measures and Jordan decomposition; the Lebesgue-Stieljes integral; the Helly convergence theorem
   • Fourier transforms: discrete, continuous, and finite; complex-analytic extension; Floquet transform
   • Finite-dimensional spectral theory: self-adjoint operators and the spectral resolution; normal operators
   • Spectral resolution for differential operators through the Fourier transform
   • Resolvent and spectral measures


2. Basic models of scattering and resonance: the generalized Fourier transform
   
   
   • Quantum: The Schrödinger equation on the line with a potential well
   • Classical: The Resonant Lamb model: An oscillator attached to a string
   • Modified, or scattered, waves
   • Eigenvalues and bound states
   • Preservation of continuous spectrum for wave scattering
   • Eigenvalues embedded in the continuous spectrum and resonance


3. Applications and open problems
   
   
   • The equations of acoustics and electromagnetics
   • Infinite periodic quantum graphs
   • Bound states in the continuum for open periodic waveguides
   • Embedded eigenvalues, resonance, and asymmetry
   • Plasmon-induced resonance and field enhancement

Course topics for spring 2019

The course will mainly focus on mathematical studies of plasmonic resonances. These include plasmon for nano particles, surface plasmons, resonance in extraordinary optical transmission through nano-holes. The mathematical studies will be mainly based on integral equations, and their asymptotic analysis and numerical solution, homogenization theory. If time allows, inverse problems arising in acoustic and electromagnetics may also be introduced.

Literature

We will not use any one source as a text book. The lectures will draw from our notes and various items of literature.

There is a bibliography of relevant works below, with links to some PDF files of excerpts.

Evaluation

Evaluation is based on participation in the course and a good-faith effort to learn the material. Fullfilling these basic requirements warrants a grade of A in the course. An A+ is reserved for students who submit excellent solutions to the exercises in the notes.

Lecture Notes and Problem Sets

Bibliography

1. 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.
2. 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.
3. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).
4. Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, Wiley 1999.
5. Maciej Zworski, Resonances in Physics and Geometry, Notices of the AMS, 1999.
6. Maciej Zworski, Mathematical Theory of Scattering Resonances Lecture notes, Version 0.1, 2016.
7. Maciej Zworski Mathematical Study of Scattering Resonances 2016.
8. 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.
9. Stephen P. Shipman and Aaron T. Welters, Resonant electromagnetic scattering in anisotropic layered media, J. Math. Phys. 54(10) 103511-1--40 (2013). pdf
10. Gayan S. Abeynanda and Stephen P. Shipman, Dynamic Resonance in the High-Q and Near-Monochromatic Regime, 2016
11. Shanhui Fan and John D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Physical Review B, Vol 65, 235112 (2002).
12. Stephen P. Shipman and Stephanos Venakides, An Exactly Solvable Model for Nonlinear Resonant Scattering, Nonlinearity, Vol. 25, No. 9 (2012) 2473-2501.
13. 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.
14. Gerald B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995. Excerpt: Chapter 0.
15. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1973.