# Seminar on Schrödinger operators

Prof. Bernd Ammann, Prof. Gunnar Bali

## Content of the Seminar

In this seminar we want to study the Schrödinger operator -Δ+V(x) as a densely defined differential operator from L^{2}(ℝ^{3}) to L^{2}(ℝ^{3}), mainly concerned with the Coulomb potential V(x)=-c/||x||. We are also interested in generalizations and in connections to representation theory. In particular, this leads to a good understanding of the orbital model in physics and chemistry.
In order to study this subject we have to start with foundations in functional analysis, with a focus on the aspects needed for Schrödinger operators. We will discuss self-adjoint extensions, discrete and essential spectra, the characterization of the essential spectrum in terms of Weyl sequences. As the potential is unbounded, eigenfunctions are in general not smooth. We show that eigenfunctions are bounded, and discuss their (higher) differentiability. Generalisations we have in mind are the multi-electron Schrödinger equations and Dirac operators with Coulomb type potentials.
The seminar will mainly follow the book by Teschl cited below, with some additional talks at the end.

## Literature

Main source
Other literature
- H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon, Schrödinger Operators (with applicatiosn to Quantum Mechanics and Global Geometry)
- Michael Reed and Barry Simon, Methods of modern mathematical physics, Vol. I-V
- Dirk Werner, Funktionalanalysis, Springer
- Hans Wilhelm Alt, Lineare Funktionalanalysis, Springer
- Siegfried Großmann, Funktionalanalysis (im Hinblick auf Anwendungen in der Physik), AULA-Verlag
- Bernd Thaller, The Dirac equation, Springer
- Gernot Münster: Quantentheorie Direct Link, available from the UR campus or with VPN
- Chris J. Isham: Lectures on Quantum Theory

Further references are given in the program.
## Time and Place

Monday, 8:15-10.00 in **M103**
## Organisational meeting

Wednesday, July 27, 16:15, in PHY 4.1.12 (Physics department)
It will be possible to follow via zoom.
## Program

The preliminary program of the seminar.
## Registration

- Come to the organisational meeting (see above).
- Please register via Email to Bernd Ammann.
- Please also register in G.R.I.P.S., as we send announcements
via this system.

## Related web pages

## Formal requirements

- For ECTS attributed by the math department: see KVV (math)
- For ECTS attributed by the physics department: ... to be clarified ...

Bernd Ammann, 17.02.2023

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