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Resonance in Wave Scattering
Vertically Integrated Research
Math 49977, Fall, 2012
Place: Lockett Hall 112
Time: Monday and Friday, 3:004:20
Office: Room 314 of Lockett Hall
Telephone: 225/5781674
Email: shipman@math.lsu.edu
Office Hours: Monday 9:0011:00, Wednesday 9:0011:00, or by appointment
Description
The phenomenon of resonance is familiar in popular and scientific tradition and commonly lies behind acoustic, electromagnetic, and mechanical processes and devices. We witness it in events such as the collapse of the Tacoma Narrows bridge, the shattering of a glass by acoustic resonance, anomalous absorption by the noble gases at specific energies, and supersensitive frequencydependence of light reflection from periodic surfaces. The topic of this course will be a mathematical theory that applies to a great variety of problems of resonance in wave scattering by objects in quantum and classical wave mechanics.
The primary literature reference will be the lecture notes on scattering resonances by Maciej Zworski. The classes will be run like a seminar, in which students and faculty will take turns presenting portions of the notes, related examples, and supporting material. Some references are given below.
The mathematical theory of resonance involves sophisticated methods of complex variables and operator theory of differential equations. Even so, there is a rich array of simple and interesting models whose analysis is accessible to undergraduate students. The role of graduate students and faculty will be to learn and present the notes and supporting material. Undergraduate students will gain exposure to advanced mathematical techniques but will not be expected to grasp all the mathematics in the notes or lectures. Their role will be to work out and present models that illuminate specific resonant scattering phenomena.
Outcomes of the course
Four groups of undergraduate students did poster and oral presentations for the Mathematics Department. Each project investigated an instance of resonance and worked out the mathematics and numerical computations.
Here are some PHOTOS of the presentations.
Polarization Rotation on a Complex Scatterer: Jeremy Baumgartner, Patrick Keiffer, and Jerome Weston.
Link to the group's Poster.
This group analyzed a mathematical model that imitates the conversion of a linearly polarized photon into a circularly polarized one when it interacts with a twolevel atom. The two levels are produced by the Zeeman effect, in which a magnetic field splits a single electronic energy level into two by imposing opposite spins. The model captured the resonance between the photon and the spin states and showed how this resonance biases the amount of right or left circular polarization imparted to the photon.
Guided Modes and Resonance in Anisotropic Media: Laura Johnson, Joe Poynot, and Andrew Rogers.
Link to the group's Poster.
Electromagnetic radiation in an ambient medium excite guided modes of a slab waveguide, resulting in resonance. Anisotropy in the ambient medium causes electrically polarized waves to propagate but prohibits the propagation of magnetically polarized waves. This allows one to trap energy as a guided mode in the material slab. In the perfectly decoupled situation, this guided mode does not interact with the propagating electrically polarized waves. But when one perturbs the material properties of the slab, coupling occurs and results in interesting resonant behavior. This group worked out the mathematics for this scattering problem starting from the Maxwell equations. Numerical computations revealed sharp anomalies in the transmission of energy across the slab when the frequency of the radiation matched that of the guided mode.
Resonance on a Line Defect in a 2D Lattice: Emelie Mativi, Tyler Meyer, Ian Runnels, and Jeremy Tillay.
Link to the group's Poster.
This group studied a different way to couple waveguide modes to radiation. Instead of altering the homogeneous material properties of the slab, one imposes a weak periodic defect. This is difficult to analyze exactly in the electromagnetic problem, so a twodimensional discrete model that exhibits the essential resonant features was devised and worked out exactly. The simplest periodicity of two lattice points is enough to allow coupling between propagating waves in the ambient lattice and guided modes of an embedded onedimensional line of defective lattice points.
Resonnce Behavior in Embedded Impurity Bands: Christina Davis, Dominique Gautreau, Charles Stephens, and Andrew Williamson.
Link to the group's Poster.
A defect in a onedimensional lattice can trap oscillatory modes at specific frequencies. When the defect is repeated a large finite number of times with sufficient separation, the trapped modes of all of the defects interact and produce a nearcontinuum of closelyspaced frequencies at which the defective region can trap energy. As the number of defects tends to infinity in both directions, an "impurity band" of frequencies admitting oscillations along the entire "doped" lattice emerges. By coupling two of these systems together to produce an infinite bilayer of coupled beads, one can arrange for the impurity band to associated with even motions of this bilayer and embed this band within the radiation spectrum of the odd motions. Perturbations of this perfect system cause resonance due to interaction between the (now almost) even impurity states and the (now almost) odd radiation modes.
Prerequisites
For graduate students: Math 7311 (real analysis) and complex variables.
For undergraduate students: Calculus. Linear algebra and complex variables are also fundamental to the subject but can be learned to the extent needed during the course of the projects.
Expectations
Show up on time, be interested, say stuff in class and out, and particpate !!!
Graduate Students will be charged with presenting material on the theory of resonance, mainly coming from the notes of M. Zworski, but also including supplementary material as needed. They may also be involved in tutoring undergraduate students and assisting in their projects.
Undergraduate students will work in groups of two to four students on a project involving some form of resonance. A typical project will investigate the way that waves propagate in a medium (disturbances of a lattice or a continuous medium or electromagnetic waves) and how they resonate at defects or cavities within the medium. Student will learn the necessary mathematics as they progress.
The projects will involve theory, literature research, and/or scientific computation and computer programming.
Students will present their work as they progress, and we will try to frame the ideas in the mathematical theory of resonance. The projects will be exposited in a written paper and on a poster to be presented to the Math Department. Students may also give an oral presentation to the department.
Schedule
Mon., Aug. 20 
Concepts of resonance and objectives of the seminar 
Fri., Aug. 24 
Participants present examples of resonance from the literature 
Mon., Aug. 27 
Undergraduate presentations of past projects on resonance (Emelie, Jeremy T.) 
Fri., Aug. 31 
Undergraduate presentations of past projects on resonance (Tyler, Andrew R.) 
Fri., Sept. 7 
Introduction to wave scattering on the line 
Sept. 10, 14 
Defects, scattering, and trapped energy in discrete and continuous systems 
Sept. 17 
Scattering for the 1D lattice

Sept. 21 
The wave equation: The Green function (Robert Viator) 
Sept. 17 
The wave equation: The resolvent (Robert Viator) 
Sept. 24 
The wave equation: Analytic continuation of the resolvent (Robert Viator) 
Sept. 28 
The wave equation: Analytic continuation of the resolvent (Robert Viator) 
Oct. 1 
Scattering for the wave equation: A local potential (Lokendra Thakur) 
Oct. 5 
Scattering for the wave equation (Lokendra Thakur) 
Oct. 8 
Project presentations 
Oct. 12 
Project presentations 
Oct. 15, 19, 22 
Continuation of scattering for the wave equation 
Fri., Oct. 26 
Discussion of project on photonatom interaction 
Mon., Oct. 29 
Complex poles 
Fri., Nov. 2 
How to do a Poster: Discussion across groups 
Mon., Nov. 5 
Scattering of a pulse (Gayan Abeynanda) 
Fri., Nov. 9 
Scattering of a pulse (Gayan Abeynanda) 
Mon., Nov. 12 
Cochlear resonance (Charles Stephens) and other stuff 
Fri., Nov. 16 
Ian Runnels' presentation and Lamb model 
Mon., Nov. 19 
The Lamb model: Spectral Theory, Fourier integrals, and the residue calculus in complex variables (Lokendra Thakur) 
Mon., Nov. 26 
TMF and EMTakeover presentations 
Fri., Nov. 30 
Presentation rehearsal: Coates 151 


Thurs., Dec. 6 
Final poster and oral presentations 
Assignments
Fri., Aug. 24 
Present an example of resonance from the literature, pointing out the essential features that resonance should exhibit, as discussed on Monday. 
Aug. 27Sept. 5 
Learn basic complex variables and vibrations in lattices: propagation, defect modes, and resonance 
Sept. 1014 
Get started on a project, in coordination with me.

Mon., Oct. 29 
Have a draft of your posters ready to show in class.

Fri., Nov. 9 
Turn in a new draft of your poster.

Fri., Nov. 30 
Poster must be done and ready to present.

Literature

David Bindel, Resonance Sensitivity for Schrödinger, Notes, 2006.

David Bindel and Maciej Zworski, Theory and Computation of Resonances in 1D Scattering (website).

J. W. Brown and R. V. Churchill, Ch. 1,
Complex Variables.

David C. Dobson, Fadil Santosa, Stephen P. Shipman, and Michael I. Weinstein,
Resonances of a Potential Well with a Thick Barrier,
SIAM J. Appl. Math., Vol. 73, No. 4 (2013) 14891512.

Maciej Zworski, Resonances in Physics and Geometry, Notices of the AMS, 1999.

Maciej Zworski, Lectures on Scattering Resonances Lecture notes, Version 0.01, 2011.
