Stephen P. Shipman
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Department of Mathematics
Louisiana State University
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Topics in Material Science: Mathematics of Resonance

Math 7384-1
Louisiana State University
Spring Semester, 2014

Prof. Stephen Shipman


Place: Lockett 111
Time: 11:30-12:20 Mon., Wed., and Fri.

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


Course Description

Resonance occurs when waves in a spatially extended system excite the vibrational modes of an oscillator. This results in field amplification in the oscillator at critical frequencies and wave scattering that is hyper-sensitive to frequency detuning. A precise analysis of resonance centers on the complex poles of the resolvent of an underlying operator, or, equivalently, the poles of a "scattering matrix". When a pole is on the real axis and is embedded in the continuous spectrum of the operator, delicate resonance phenomena are revealed through perturbation analysis of the scattering matrix. The mathematics of resonance intimately connects complex analysis, spectral theory, Fourier analysis, and perturbation of linear operators.

Prerequisite

The prerequisites are real and complex analysis. Spectral theory and some PDEs will also be useful.

Course Topics

  1. The simplest oscillators in excessive detail
    complex first-order harmonic oscillator
    real second-order harmonic oscillator
    the free Schrödinger equation on the line
    the wave equation on the line
  2. Lamb model
    physical approach
    in theoretical context
    the Schrödinger version
  3. Elaborated Lamb model with embedded eigenvalue
    full theory---Schrödinger and wave versions
    questions concerning time dynamics and small parameters
  4. Resonance in a discrete model---discrete version of Zworski's notes
    analytic continuation of the resolvent
    expansions in normal modes
  5. Resonance according to Zworski's notes
    analytic continuation of the resolvent
    expansions in normal modes
  6. Perturbation of systems with embedded eigenvalues
    theory of scattering resonances and anomalies according to Shipman/Venakides/Welters
    various applications, Fano resonance, etc.
  7. Extension to higher dimension; the Helmholtz resonator
  8. Periodic structures with defects
    Floquet theory
    bound states at a defect and embedded eigenvalues
  9. Lossy systems
    Extensions of lossy systems to conservative systems
    The Naimark theorem
    The unique-extension theorem of Figotin and Schenker

Literature

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.

Assignments

I may assign problems periodically.

Evaluation

Evaluation of performance in the course is based on any assignments or other activities such as presentations.

Lecture Notes

Notes on Resonance

Bibliography

  1. David Bindel, Resonance Sensitivity for Schrödinger, Notes, 2006.
  2. David Bindel and Maciej Zworski, Theory and Computation of Resonances in 1D Scattering (website).
  3. Maciej Zworski, Resonances in Physics and Geometry, Notices of the AMS, 1999.
  4. Maciej Zworski, Lectures on Scattering Resonances Lecture notes, Version 0.01, 2011.
  5. 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.
  6. Stephen P. Shipman and Aaron T. Welters, Resonant electromagnetic scattering in anisotropic layered media, J. Math. Phys. 54(10) 103511-1--40 (2013). pdf
  7. 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.
  8. A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, §36: The Stieljes integral, 1968 (translation by Silverman, Dover 1975).
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