Kaisa_2012_3_photo by Veikko Somerpuro

Spectral theory of infinite dimensional operators and its applications

Please check regularly the Messages section below and the Moodle Forums available via the link below for information about how the course will be adjusted to the exceptional measures in period IV.

## Jani Lukkarinen

Published, 23.4.2020 at 16:12

Final project topics are now available below. There was also a last minute relaxation of the grading criteria to Exercise 12.1: see Homework 12 below.

## Jani Lukkarinen

Published, 21.4.2020 at 15:20

New hints added to Exercise sheet 12 below.

## Jani Lukkarinen

Published, 24.3.2020 at 17:55

New hint added to Exercise 9.4.

## Jani Lukkarinen

Published, 21.3.2020 at 15:25

The first Zoom session of the course will be on Tuesday March 24 starting at 14:15, and the Zoom address will be send to you by e-mail. Before the session, please study Sections 5.2 and 5.3 from the textbook by Davies, as well as the related Notes which can found in the Materials section below.

## Jani Lukkarinen

Published, 18.3.2020 at 16:48

For the proof of the spectral theorem missing from the book of Davies (Theorem 5.4.1) I will use the material from Chapter 3 of a book by Teschl. A link to a related web page has been added below: you can prepare by checking that you can find the correct PDF file (2nd edition) already now.

## Jani Lukkarinen

Published, 18.3.2020 at 16:13

Several new discussion forums opened in the new Moodle area of the course. Follow the link above to participate.

## Aleksis Vuoksenmaa

Published, 18.3.2020 at 14:48

Here is some specific information about the remaining exercises.

Each week (starting from this week, i.e. set 8), the deadline for turning in the exercises is on FRIDAY at 10 am. Please send your solutions to me by email (aleksis.vuoksenmaa@helsinki.fi).

I will grade each exercise on a scale 0-3 using the following criteria:
- 0 = no answer
- 1 = decent effort
- 2 = mostly correct, but lacks an important step or contains a notable error
- 3 = In essence, a correct and complete answer.

I will return the marked solution to you by the next Monday, and you will have time to correct your answers (if needed) until the following Friday. Therefore, for each problem set you have a second chance to raise your weekly points.

For example, if you return your solutions to this weeks exercises on Friday 20.3., I will evaluate and comment them by Monday 23.3. If you get, say, 2/3 points from problem 8.1, you have until Friday 27.3. (10 am) to send your revised solution to 8.1 to me. If your revised solution matches the criteria for 3 points, I will give you 3/3.

This second round of returning the solution never lowers your points.

Note that on each Friday, there are two deadlines:
1) Your solutions for this week's exercises
2) Possibly your revised solutions for last week's exercises

Additionally, each Thursday from 10:15 to 11:45 we have a Zoom session, where we can discuss the current exercises. These are not mandatory, but please join if you are stuck with some exercises, or want to ask clarifying questions. Tomorrow, you can ask questions about problem set 8.

If you have any questions, please don't hesitate to contact me (aleksis.vuoksenmaa@helsinki.fi).

## Jani Lukkarinen

Published, 16.3.2020 at 11:07

The lectures on Monday March 16 and Tuesday March 17 have been *cancelled*. All teaching of the course will gradually be moved to digital format: more details to follow by e-mail over this week.

## Jani Lukkarinen

Published, 12.3.2020 at 10:45

Exercise session on 12.3. has been cancelled. If you want the credits from the session (HW7), please send your solutions to Aleksis by email (aleksis.vuoksenmaa@helsinki.fi). You can scan or take pictures of the handwritten solutions, for example, or compile a pdf-file: please contact Aleksis for further details.

The deadline for the solutions is Sunday 15.3.2020 at 16:00. In case you want to discuss these exercises in person, you can find Aleksis in Ratkomo (Exactum, 3rd floor) on Monday 16.3. between 10:15 and 11:45.

## Jani Lukkarinen

Published, 2.3.2020 at 11:56

The first partial exam will be held on Wednesday, March 4th, at 13:15-16:00 in Exactum, room C122. The exam covers lecture and exercise material from the first period: Chapters 1.1-1.2, 1.4-1.5, 4.1-4.2 from the course textbook by Davies, the PDF notes attached below, and Exercises 0-6. In addition to standard writing equipment, you are allowed to bring in and consult one *handwritten* two-sided A4 sheet of personal notes during the exam.

## Pages

### Timetable

Here is the course’s teaching schedule. Check the description for possible other schedules.

DateTimeLocation
Mon 13.1.2020
12:15 - 14:00
Tue 14.1.2020
12:15 - 14:00
Tue 14.1.2020
14:15 - 16:00
Mon 20.1.2020
12:15 - 14:00
Mon 27.1.2020
12:15 - 14:00
Tue 28.1.2020
14:15 - 16:00
Mon 3.2.2020
12:15 - 14:00
Tue 4.2.2020
14:15 - 16:00
Mon 10.2.2020
12:15 - 14:00
Tue 11.2.2020
14:15 - 16:00
Mon 17.2.2020
12:15 - 14:00
Tue 18.2.2020
14:15 - 16:00
Mon 24.2.2020
12:15 - 14:00
Tue 25.2.2020
14:15 - 16:00
Mon 9.3.2020
12:15 - 14:00
Tue 10.3.2020
14:15 - 16:00
Mon 16.3.2020
12:15 - 14:00
Tue 17.3.2020
14:15 - 16:00
Mon 23.3.2020
12:15 - 14:00
Tue 24.3.2020
14:15 - 16:00
Mon 30.3.2020
12:15 - 14:00
Tue 31.3.2020
14:15 - 16:00
Mon 6.4.2020
12:15 - 14:00
Tue 7.4.2020
14:15 - 16:00
Mon 20.4.2020
12:15 - 14:00
Tue 21.4.2020
14:15 - 16:00
Mon 27.4.2020
12:15 - 14:00
Tue 28.4.2020
14:15 - 16:00

### Other teaching

23.01. - 27.02.2020 Thu 10.15-12.00
12.03. - 02.04.2020 Thu 10.15-12.00
16.04. - 30.04.2020 Thu 10.15-12.00
Jani Lukkarinen
Teaching language: English

### Material

Notes about the lectured material will be added here during the course.

## Lecture material

#### Final project work

Please choose one of the project topics listed below, and inform the lecturer of your choice by April 29th: further details are provided in the beginning of the PDF file. In particular, there is a possibility for individual discussion session, as explained in the file.

#### Homework sets, second period

Related material: Section 4.2
[Please prepare your solutions in written format and send them to Aleksis by e-mail]
Related material: Theorem 4.2.13, Sections 5.1 and 5.2
[Added hint to Exercise 4 on 24.3.]
Related material: Sections 5.1, 5.2 and 5.3
[clarifed assumptions in Exercise 1]
Related material: Teschl, Secs. 1.4, 2.2, 2.5, as detailed in the Notes in Materials section
Related material: Teschl, sec. 3.1
[23.4.: grading of Problem 1 has been relaxed]
Related material: Sections 5.4-5.6 from Davies

#### Homework sets, first period

(Ex tempore -session)
Related material: Introduction and Sections 1.1-1.2.10. of Davies (2007)
Related material: Section 1.2.
Related material: Section 1.2 and the notes about integration
Related material: Theorem 1.3.4 and Section 1.4
Related material: Sections 1.4 and 1.5
Related material: Chapter 4 until Theorem 4.2.4

### Description

Optional course.

Master's Programme in Mathematics and Statistics is responsible for the course.

The course belongs to the Mathematics and Applied mathematics module.

The course is available to students from other degree programmes.

B.Sc.-level mathematics, Functional analysis

Sobolev space theory, Fourier analysis, theory of distributions

Knowledge of basic spectral theory for bounded and unbounded operators in Hilbert spaces and applications to partial differential equations

Recommended time/stage of studies for completion: 1. or 2. year

Term/teaching period when the course will be offered: varying

Unbounded operators in Hilbert spaces; closed, symmetric and self-adjoint operators; spectral theorem; perturbation theory; applications to elliptic PDE

Required: lecture notes.

Recommended: Reed-Simon, Methods of modern mathematical physics; Davies: Spectral theory and differential operators

Lectures and exercise classes

Exam and excercises, Course will be graded with grades 1-5

Exam, other methods will be described later