Math 33304
Applied Analysis
The Applied Analysis sequence (I, II and III) is aimed at graduate students who
have completed the Analysis Sequence. The course covers topics from Mathematical
Physics and is taught in self-contained modules. For this quarter we will focus
on both basic and recent results for the nonlinear Schroedinger equation (NLS):
- Schroedinger Eq as a model: in Quantum Mechanics via stationary
phase method; in Bose-Einstein condensates; in Optics via multiple scale analysis
and paraxial approximation
- Analysis of Linear Schroedinger Eq, semi-group of operators, Stricharz estimates,
L^2_{loc} estimates,wave operators
- Local and global existence for NLS
- Existence of bound states in NLS via variational methods and via
bifurcation from linear ones
- Orbital stability/instability of bound states in NLS variational
methods and concentration compactness, sharp interpolation estimates and
blow-up. Asymptotic stability via center manifold technique
- Effect of time dependent perturbations, resonances with radiation