Introduction to scattering theory for Schrödinger equations
(graduate course, SISSA -- May-June 2016)


Lecturer: Kenji Yajima (Tokyo Gakushuin)

venue and schedule: Monday 14:30-16:00 A-136 - Wednesday 14:30-16:00 A-134
start: Friday 6 May 2016, 11:15, A-136
end: Wednesday 29 June 2016
duration: 40 hours (2 cycles)


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 smoothing 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


 Pre-requisites: a general knowledge at the undegrad level of:
  • Introductory functional analysis
  • Spectral representaion theorem for selfadjoint operators
  • Introductory Fourier analysis
  • Lp spaces, Sobolev spaces
  • Integration of vector valued functions.
  • Fourier transform of vector valued functions.
These pre-requisites will be reviewed also in the courses "Mathematical Quantum Mechanics" or "Introduction to Spectral Geometry" that precede this course.


Literature: discussed in class










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