授業の目的 【日本語】
Goals of the Course(JPN) | | |
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授業の目的 【英語】
Goals of the Course | | Title : Operator theory in Hilbert spaces
This course will provide the basic notions of operators on Hilbert spaces: bounded and unbounded operators,
self-adjoint operators, the spectral decomposition of self-adjoint operators. These notions will be illustrated with some examples of operators appearing in quantum mechanics or in PDE. A powerful tool of functional analysis, namely the commutator methods, will then be introduced and studied. Typical applications of these methods for spectral and scattering theory will finally be presented |
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到達目標 【日本語】
Objectives of the Course(JPN)) | | |
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到達目標 【英語】
Objectives of the Course | | | Develop and understand some standard techniques of Hilbert space theory, and in particular the spectral theory. For scattering theory, understand its role as a comparison theory and develop the necessary tools for applying this theory in various contexts (from dynamical systems to group representations). For Mourre theory (commutator methods), get the main idea underlying this powerful tool. |
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授業の内容や構成
Course Content / Plan | | Content:
1) Hilbert space and bounded linear operators
2) Unbounded operators
3) Examples
4) Spectral theory for self-adjoint operators
5) Scattering theory
6) Commutator methods |
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履修条件
Course Prerequisites | | | Knowledge on standard undergraduate linear algebra, calculus and advanced calculus. |
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関連する科目
Related Courses | | | Any courses on functional analysis. |
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成績評価の方法と基準
Course Evaluation Method and Criteria | | | Grades based on attendance and on written reports. An active participation of the students is expected. |
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教科書・テキスト
Textbook | | |
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参考書
Reference Book | | W. Amrein, Hilbert space methods in quantum mechanics, EPFL press, 2009.
W. Amrein, A. Boutet de Monvel, V. Georgescu, Co-groups, commutator methods and spectral theory of N-body Hamiltonians, Birkhauser, 1996.
T. Kato, Perturbation theory for linear operators, Springer, 1995.
G. Teschl, Mathematical methods in quantum mechanics, with applications to Schroedinger operators, AMS, 2009.
D. Yafaev, Mathematical scattering theory: general theory, AMS, 1992. |
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課外学習等（授業時間外学習の指示）
Study Load(Self-directed Learning Outside Course Hours) | | | Students are supposed to read their notes between two lectures. The subjects of the reports can be chosen according to the interest of the students |
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注意事項
Notice for Students | | Additional information is available on
http://www.math.nagoya-u.ac.jp/~richard/Operators2026.html |
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他学科聴講の可否
Propriety of Other department student's attendance | | |
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他学科聴講の条件
Conditions of Other department student's attendance | | | This course is open for any students at Nagoya University. Motivated undergraduate students are also welcome. |
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レベル
Level | | |
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キーワード
Keyword | | | Spectral and scattering theory, unitary operators |
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履修の際のアドバイス
Advice | | | It is certainly easier to attend a course in English and to discuss in this language in a quiet university environment rather than later in a busy life. |
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授業開講形態等
Lecture format, etc. | | | Face-to-face in room 309 of the math building |
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遠隔授業（オンデマンド型）で行う場合の追加措置
Additional measures for remote class (on-demand class) | | |
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