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Linear and Non-Linear Schrodinger Equations: Well-Posedness, Scattering

External Lecturer: 
Kenji Yajima
Course Type: 
PhD Course
Academic Year: 
2015-2016
Period: 
May - June 2016
Duration: 
40 h
Description: 

Webpage:

http://people.sissa.it/~alemiche/teaching2016_Yajima.html

Topics:

1. Free Schrödinger equation. 1.1. Free propagator, MDFM-formula,
1.2. Large time behavior, Asymptotic expansion
1.3. Lp-Lq estimates, Strichartz’ estimate I
1.4. Strichartz’ estimate II
1.5. Application of Strichartz’ estimates to NLSE

2. Free Schrödinger operators 2.1. Spectral decomposition I
2.2. Spectral decomposition II
2.3. Limitting absorption principle (LAP)
2.4. Kato smoothness and local smoothin property
2.5. Resolvent kernel

3. Self-adjointness 3.1. Kato-Rellich theorem and selfadjointness of electronic systems.
3.2. Kato’s inequality and positive potentials.
3.3. quadratic forms and Kato potentials
3.4. Diamagnetic inequality
3.5. Leinfelder-Simader’s theorem
3.6. Essential spectrum and discrete spectrum

4. One-body scattering theory 4.1. RAGE theorem
4.2. Existence of wave operators
4.3. Asymptotic completeness
4.4. Proof via Enss method
4.5. Stationary scattering theory a. Limitting absorption principle , proof by Agmon-Kuroda
b. Mourre theory and proof of LAP by Mourre theorem
c. Proof of asymptotic completeness via LAP
d. Eigenfunction expansions via scatterin solutions
e. Wave operators as transplantations
f. Scattering amplitute and scattering matrix 

Prerequisites: 
Basic knowledge of linear algebra and functional analysis.
Further pre-requisites will be given in the courses "Mathematical Quantum Mechanics" or "Introduction to Spectral Geometry" that precede this course.
Location: 
Mondays A-136, Wednesdays A-134, Friday A-136

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