Hilbert space methods for quantum mechanics

Course Overview

This course will provide an overview of some classical tools of functional analysis which have been partially developed for quantum mechanics. In particular, an introduction to spectral and scattering theory will be presented. These theories have deep connection with other branches of mathematics, like PDE, operator algebras or dynamical systems. Some up-to-date tools of spectral theory will also be introduced, as for example commutator methods for spectral theory. In order to provide a large panorama on the subject together with applications, some details might be omitted, but references for all proofs will be provided.

References

The two main references for this course will be

• Amrein: Hilbert space methods in quantum mechanics, 2009
• Teschl: Mathematical methods in quantum mechanics, 2009

Plan of the course

Tentative program:

1. Hilbert space and bounded operators,
2. Unbounded operators,
3. Self-adjoint operators and spectral theory,
4. Some examples,
5. Evolution group and scattering theory,
6. Commutator methods.

Keywords

Hilbert space, self-adjoint operators, spectral and scattering theory, commutator methods.

Required Knowledge

Knowledge on standard undergraduate functional analysis.

Attendance

This course is open for any students at Nagoya University as one of the “open subjects” of general education.

Method of Evaluation

Grades based on attendance, a written report, or an examination.

Lecture Notes

full document (100 pages)

• Hilbert space methods for quantum mechanics

Lecture Notes

Title

Contents

Chapter 1

• Hilbert space and bounded linear operators

Chapter 2

• Unbounded operators

Chapter 3

• Examples

Chapter 4

• Spectral theory for self-adjoint operators

Chapter 5

• Scattering theory

Chapter 6

• Commutator methods

Bibliography

• Bibliography

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