### Hilbert space methods (Summer Semester 2020)

- Lecturer: JProf. Dr. Xian Liao, Dr. Lucrezia Cossetti
- Classes: Seminar (0173500)
- Weekly hours: 2

### Time

The seminar is scheduled at 15:45 on Wednesday.

### Place

**This is a online Seminar. We are going to "meet" online on the Microsoft Team platform. Anyone with a KIT account should have the permission to use this platform and more information can be found here
http://www.scc.kit.edu/en/services/ms-teams.php
**

**We have created a team named "IANA_Seminar_HibertSpaceMethod". Please click the following link to join this team when you can get access to the MS Teams
https://teams.microsoft.com/l/channel/19%3a0e8cc9fdf04b4e5d8130e5f48b829045%40thread.tacv2/General?groupId=8f7aacf5-faf7-4112-8c8d-792d16d423a0&tenantId=4f5eec75-46fd-43f8-8d24-62bebd9771e5
You are welcome to come to this team and to ask related questions.**

### Registration

**The first meeting on Tuesday, 17.03.2020 is CANCELLED. Please send us a email as soon as possible for the registration of the seminar. We will distribute the topics after the reception of the registrations. DEADLINE: 10.04.2020.**

### Abstract

This seminar is concerned with the Hilbert space aspects of quantum mechanics, in a nutshell it covers the classical Hilbert space theory, the measure theory, the operator theory and it eventually provides a first introduction to scattering theory.

It can also be seen as a deepening/natural extension of the (just concluded) Functional Analysis course (FA) which has been held in the passing winter semester and as a preview of the Spectral Theory course which will be held in the upcoming summer semester.

The first part of the seminar (see 1-3 below) intends to recover some basic concepts on Hilbert spaces, measure theory and linear operators (the majority of which has been already presented in the aforementioned FA course) which will be used for the arguments treated later in the seminar. This makes (at least) the first part of the course totally accessible to undergraduate students.

In the second part (4-10) we aim at investigating more deeply the fundamental notion of self-adjointness of (unbounded) linear operators. A detailed study of the relation between symmetric and self-adjoint operators is in the purpose of the seminar where a particular attention is given to the criteria ensuring self-adjointness of symmetric operators. The invariance of self-adjointness under different classes of perturbation is also object of interest. Some criteria will be seen at work in the particularly interesting case of Schrödinger operators.

The spectral theory of self-adjoint operators and more specifically the precise characterisations of the spectral parts of self-adjoint operators will be mentioned.

The cornerstone result for self-adjoint operators represented by the spectral theorem will be also treated and a proof will be sketched.

In the final part of the seminar’s program (11-14), the material presented so far and the mathematical tools acquired (in a purely stationary context) will be applied to better understand the evolution of states when the dynamic is generated by a perturbed self-adjoint operator and it is investigated how the evolution deviates from the free one. More specifically, we are interested in introducing the main aspects of the scattering theory in a Hilbert space framework, namely underlying the differences between scattering and bound states and investigating both the problems of existence and asymptotic completeness of the wave operator.

### Instructors

Schedule | |||
---|---|---|---|

Seminar: | Wednesday 15:45-17:15 | online | Begin: 20.4.2020, End: 20.7.2020 |

### Workload Description (14 meetings)

Here it will follow a description of the single meeting workload:

- Definition and elementary properties of Hilbert spaces. Dual of a Hilbert space and Riesz lemma. Measure theory and integration in the sense of Lebesgue (basic notions). (
**A**, Chapter 1)*assigned to Alexandru Ionita*Notes_Ionita - Linear operators: bounded operators, projections and isometries and compact operators. (
**A**, Sect. 2.1-2.3) Notes_Liao_Topics2-3 - Linear operators: unbounded operators, adjoint operator, symmetric and (essentially) self-adjoint operators, multiplication operators. (
**A**, Sect. 2.4-2.5) - Resolvent and spectrum of closed operators. Perturbation of self-adjoint operators: relatively boundedness and Kato-Rellich theorem, relatively compactness. Example: Schrödinger operators. (
**A**, Sect. 2.6-2.7) Notes_Cossetti_Topics4-5 - Symmetric extensions of symmetric operators: deficiency indices, the method of the Cayley transform, von Neumann Formula. (
**A**, Sect. 3.1) - Differential operators with constant coefficients. Minimal and maximal operators. (
**A**, Sect. 3.2) Notes_Liao_Topics6-7 - Differential operators with constant coefficients. Case study: Schrödinger operators.(
**A**, Sect. 3.3) - Spectral parts of self-adjoint operators: Stieltijes measures. Spectral measures. (
**A**, Sect. 4.1-4.2)*assigned to Mats Hansen*Notes_Hansen_StieltijesMeasure Notes_Hansen_SpectralMeasure - Spectral parts of self-adjoint operators: definitions and characterisations. (
**A**, Sect. 4.3) Notes_Goffi_Topics 9-10 - Spectral theorem. (
**A**, Sect. 4.4) - Scattering theory: Evolution groups. Stone’s theorem. Scattering and bound states (
**A**, Sect. 5.1-5.2) Notes_Liao_Topics 11-12 - Scattering theory: Existence of Wave operator. Cook criterion. Scattering operator. Examples. (
**A**, Sect. 5.3-5.4) - Scattering theory: Asymptotic completeness. (
**A**, Sect. 5.5) Notes_Cossetti_Topics 13-14 - Scattering theory: Coulomb scattering. Modified wave operator. (
**A**, Sect. 5.8)

### Literatur

W. **A**mrein -- "Hilbert space methods in quantum mechanics", EPFL Press (hardback), ISBN 978-1-200-6681-4, pp.395 (2009)