# Lecture course: Mathematical quantum mechanics

## Tue, 12 – 14 in B 005, Thu 12 – 14 in B 006

**Problem solving classes: Dr. Jean-Claude Cuenin**

Problem sheets and further information

Problem sheets and further information

**21.02.18**The results of the exam can be looked up here. Inspection of exams: Thursday, 1 March 2018 from 11:00 until 12:00 in B 336. Certificates can be picked up in Mrs Warlimont's office from 22 February on.**07.02.18**The proof of Theorem 4.20 (a) has been streamlined in the script by considering states from the Sobolev space*H*^{2}rightaway.**12.01.18**The proof of claim (ii) in the proof of Perry's estimate (Theorem 3.18) has been made more transparent in the script and its adaptation to the situation in Remark 3.19 is worked out in more details there.**17.10.17**Prerequisites on measures and integration, Banach and Hilbert spaces and bounded operators are summarised in this handout.

**Synopsis**

The course introduces the basic elements of mathematical quantum mechanics and the necessary analytical tools. Topics to be covered include: observables as self-adjoint operators, spectral theorem for self-adjoint operators, relation between spectral types and dynamics, elements of scattering theory, many-particle systems.

**Prerequisites**

Basics of functional analysis and quantum mechanics are helpful.

**Audience**

Master students.

**Literature**

- M. Reed, B. Simon,
*Methods in modern mathematical physics, vol. I – IV*, Academic Press, San Diego - G. Teschl,
*Mathematical methods in quantum mechanics*, 2nd ed., Amer. Math. Soc., Providence, RI, 2009 - W. Thirring,
*Quantum mathematical physics*, Springer, Wien, 2002 - J. Weidmann,
*Linear operators in Hilbert spaces*, Springer, New York, 1976

[There exists an updated and largely expanded German edition in two volumes] - for a thorough physics background: A. Galindo, P. Pascual,
*Quantum Mechanics I and II*, 2nd ed., Springer, Berlin, 1989

**Contents**

0. Introduction

0.1. Why mathematical quantum mechanics?

0.2. Comparison classical mechanics and quantum mechanics

1. Unbounded operators on Hilbert space

1.1. Closed operators and adjoints

1.2. The spectral theorem for self-adjoint operators

1.3. Decompositions of the spectrum

2. Spectral types and quantum dynamics

Excursus: Compact and trace-class operators

Excursus: Operator perturbations

3. Scattering theory

3.1. Wave operators: existence

3.2. Incoming and outgoing states

3.3. Wave operators: asymptotic completeness

4. Atomic Schrödinger operators

4.1. Operators on product spaces

4.2. The HVZ theorem

*(I am grateful if you report misprints and errors in order to improve future versions.)*

**Notes of Robert Helling**

C*-algebras

Quantum states

Quantum channels2