Applications of Spectral Theory in the Material Sciences

Math 7390-2
Louisiana State University
Spring Semester, 2008

Place: Room 218 of Prescott Hall
Time: Monday, Wednesday, and Friday from 9:40 to 10:30

Instructor: Prof. Stephen Shipman
Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@math.lsu.edu
Office Hours: Monday, Wednesday, and Friday 10:40--12:00 or by appointment

Course Synopsis

The spectrum of an operator is one of the most fundamental objects that arises in problems of physics. It describes, for example, fundamental modes of vibration or fundamental shapes that are preserved through time in the dynamics of material structures. The aim of the course is a development of the role of Hilbert space in mathematical physics and especially the spectral decomposition of operators in Hilbert space, most of which are unbounded.

The syllabus is rather ambitious. I hope to delve substantially into the first four applications listed below and at least touch on the fifth. These applications represent only a tiny but important selection of problems in mathematical physics, and they will serve to build a foundation in the methods of Hilbert space and spectral analysis.

We will begin with the theory of self-adjoint operators in Hilbert space and learn the spectral theorem. We will not learn the proof of the spectral theorem, as this would require a course in itself, but rather learn what is means, how it is manifest in examples, and how it is useful in mathematical physics. We will treat several theorems in functional analysis in this manner, taking them for granted but then building rigorously upon them.

• Review of Hilbert space and some results of functional analysis.
• Review of the spectral decomposition for transformations in finite-dimensional inner-product spaces. We will pursue an exposition that leads to a generalization of the spectral decomposition for normal operators in infinite-dimensional Hilbert space, known as the "spectral theorem".
• The spectral theorem says that every normal operator in a Hilbert space is unitarily equivalent to a multiplication operator on an L² space. We will discuss the main steps in a proof of the spectral theorem, referring to the literature for the details.
• Stone's Theorem. This theorem associates with each self-adjoint operator A in a Hilbert space a unique norm-preserving dynamics through the exponential function e^-iAt, which is the unitary semigroup associated to A. This semigroup gives unique solutions to the Cauchy problem dx/dt = -iAx.
• Special classes of operators and their spectrum. The most important for us will be compact operators and multiplication operators.
• Embedded eigenvalues and resonance.
• The Fourier transform, discrete and continuous, finite and infinite. Spectral representation of the Laplacian in all these cases.

Applications.

• Quantum mechanics. The Schrödinger operator on the line with a well potential. The Schrödinger operator in 3-space with a 1/r potential well as a model of the hydrogen atom. The helium atom and associated concepts of embedded eigenvalues and resonance.
• The Laplacian and more general elliptic operators on bounded domains, which applies to such problems as diffusion (as heat), electric conduction, linear waves and vibration, electrostatics, and incompressible fluid flow. The mathematical ideas include the weak formulation of PDEs and the form domain of operators, the resolvent of operators, and the spectrum of compact operators.
• The Laplacian on the interval [0,1]. The beautiful correspondence between self-adjoint extensions of the symmetric Laplacian to the vibrating string (or diffusion in a rod) by means of boundary values is probably the best illustration of the drastic effect that the subtle notion of the domain of an operator has on the associated dynamics, or semigroup.
• Photonic crystals. We will trudge through the review article of Peter Kuchment (2001). This is a difficult article and will require students to read portions of the literature referenced therein and present expositions thereof to the class.
• Effective conductivity of micro-composite materials. The paper of Papanicolaou and Golden (1983) deals with the bulk conductivity properties of micro-structured composites in the framework of the spectral theory.

Assignments

Evaluation

Lecture Notes

Literature

1. Excerpts on compact operators:
   Naylor and Sell
   Kress
   Reed and Simon
   Riesz and Sz.-Nagy
   Akhiezer and Glazman, Chapter V; see the link below.
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. N. Dunford and J. T. Schwartz, Linear Operators: Part I General Theory, Part II Spectral Theory, Part III Spectral Operators, Wiley Classics Library, 1963.
4. Gerald B. Folland, Introduction to Partial Differential Equations Chapter 0, Second Edition, Princeton University Press, 1995.
5. Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence, Linear Algebra, Prentice Hall, 1979, 1989, 1997.
6. P. D. Hislop and I. M. Sigal, Introduction to Spectral Theory with Applications to Schrödinger Operators, Springer series in Applied Mathematical Sciences, Vol. 113, 1996.
7. K. Golden and G. Papanicolaou, Bounds for Effective Parameters of Heterogeneous Media by Analytic Continuation, Commun. Math. Phys., Vol. 80, 473-491 (1983).
8. Stephen J. Gustafson and Israel Michael Sigal, Mathematical Concepts of Quantum Mechanics, Universitext series of Springer-Verlag, 2000.
9. John D. Joannopoulos, Robert D. Meade, and Joshua N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, 1995.
10. Rainer Kress, Linear Integral Equations, Second Edition, Vol. 82 in Applied Mathematical Sciences, Springer Verlag, 1999.
11. Peter Kuchment, The Mathematics of Photonic Crystals, Chapter 7 in the volume Mathematical Modeling in Optical Science of the SIAM series Frontiers in Applied Mathematics, 207-272 (2001).
12. Arch W. Naylor and George R. Sell, Linear Operator Theory in Engineering and Science, Springer series on Applied Mathematical Sciences, Vol. 40, Springer Verlag, 1982.
13. 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, Academic Press, 1980.
14. Frigyes Riesz and Béla Sz.-Nagy, Functional Analysis, F. Ungar Publishing Co., New York, 1955 or Dover Publications, Inc., 1990.
15. Walter Rudin, Real and Complex Analysis, Third edition, McGraw-Hill, 1987.
16. Walter Rudin, Functional Analysis, Second Edition, McGraw-Hill, 1991.