2022 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on current topics in Mathematics F
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Haruya Mizutani / Yoshiyuki Kagei
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive (本館2階数学系201セミナー室)
- Class
- -
- Course Code
- MTH.E636
- Number of credits
- 200
- Course offered
- 2022
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This is an introductory course of mathematical study of the quantum scattering theory, which studies the asymptotic behaviors of solutions to some PDEs describing the motion of microscopic particles such as Schroedinger or Dirac equations. The first half of the course will be devoted to explaining the spectral theory for Self-adjoint operators and the existence and asymptotic completeness of wave operators for short-range Schrodinger equations, while the latter half part will be concerned with the boundedness of wave operators on Lebesgue spaces as an application of scattering theory to PDEs.
The goal of the lecture is to learn basic knowledge and techniques of analysis in the mathematical study of quantum scattering theory.
Course description and aims
To learn some basic techniques of Functional analysis in the spectral theory.
To understand the basic framework of mathematical quantum scattering theory.
Keywords
Quantum scattering theory; Spectral theory; Self-adjoint operator; Schroedinger equation; Wave operator; Asymptotic completeness
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be discussed (as long as time allows): ・Basic framework of Quantum scattering theory ・Spectrum of self-adjoint operators and RAGE theorem ・Limiting absorption principle for the resolvent and Mourre theory ・Smooth perturbation theory ・Existence and asymptotic completeness of wave operators for short-range Schroedinger equations ・Boundedness of wave operators on Lebesgue spaces | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None required
Reference books, course materials, etc.
黒田成俊「スペクトル理論II」岩波書店
中村周「量子力学のスペクトル理論」共立出版
谷島賢二「シュレーディンガー方程式 I, II」朝倉書店
磯崎洋「多体シュレーディンガー方程式」丸善出版
Other references will be announced in the lecture.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C351 ： Functional Analysis
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites
None