KEY COURSE TOPICS
0. OVERVIEW
I. STURM-LIOUVILLE THEORY
Sturm-Liouville systems:
regular, periodic, and singular
Eigenvalues and
eigenfunctions via phase analysis
II. INFINITE
DIMENSIONAL VECTOR SPACES
Fourier series
Dirichelet kernel
Fourier's theorem on a finite domain
Sequences leading to the Dirac delta function
Fourier transform representation
Change of basis in Hilbert space:
Orthonormal wavelet and wavepacket representations
IV. GREEN'S FUNCTION THEORY: INHOMOGENEOUS DIFFERENTIAL EQUATIONS
Homogeneous sytems
Adjoint systems
Inhomogeneous systems
The concept of a Green's function
Solution via Green's function
Integral equation of a linear system via its Green's function
Classification of integral equations
The Fredholm alternative
Green's function and the resolvent of the operator of a system
Eigenfunctions and eigenvalues via residue calculus
Branches, branch cuts, and Riemann sheets
Singularity structure of the resolvent of a system:
Poles and branch cuts
Effect of boundary conditions and domain size
V. THEORY OF SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS
IN TWO AND THREE DIMENSIONS
Partial differential equations: hyperbolic, parabolic, and
elliptic
The Helmholtz equation and its solutions in the Euclidean plane.
Geometry of the space of solutions
Plane waves vs cylinder waves:
Why, and when to use them
Sommerfeld's integral representation
Hankel, Bessel, and Neumann waves
Change of basis in the space of solutions: partial waves
Displaced cylinder waves
The cylindrical addition theorem
Method of steepest descent and stationary phase
Analytic behaviour of cylinder waves
Interior (cavity) and exterior (scattering) boundary value
problems
Spherical waves: symmetric and nonsymmetric
Cauchy problem and characteristics (time permitting)
Texts: (1) U.H. Gerlach: Linear
Mathematics in Infinite Dimensions (Chapter
1,2,3,4,5)
(2)
F.W. Byron and R.W. Fuller: Mathematics of Classical
and Quantum Physics