Math 615 — Mathematical Methods for Physics

Fall 2002

Instructor: Dan Barbasch

Time: MWF 12:20-1:10

Room: Malott 206

Prerequisites:

• A knowledge of complex variable theory up to and including evaluation of integrals using the residue theorem.
• Finite dimensional vector spaces up to and including diagonalization of hermitian matrices.
• Fourier Series
• Separation of variables of partial differential equations.

Grading:

Exams: There will be a Midterm exam at the end of October.

Homework: The homework is very important. It defines what the material of the course is. There will be a homework assignment for every week.

Final: The final covers all the material in the course equally and is given during regular exam period.

Syllabus for 615

1. Hilbert Spaces

a. Definitions and Completeness

b. Bessel's inequality

c. Orthonormal bases

d. Parseval`s equality

2. Generalized Functions (following notes by R. Strichartz)

a. Basic properties and examples

b. Differentiation of generalized functions

c. Fourier transforms

d. Applications to PDE's

e. Green's functions

3. Ordinary Differential Equations

a. Green's functions for boundary value problems

b. Eigenfunction expansions

c. Completeness of eigenfunctions (continued in 616)

4. Asymptotic Expansions

a. Definition of order of growth and asymptotic expansions

b. Method of integration by parts

c. Laplace's method

d. Method of stationary phase

e. Method of steepest descent (following reference of Bender and Orszag)

4. Optional: If time permits we will cover some topics on Lie algebras and quantum groups.