Spectral theory, fall 2016

Last modified by taskinen@helsinki_fi on 2024/03/27 10:51

Spectral theory, fall 2016

News

The second course exam will be on Thursday Dec. 15th at 14.15-16.00 in C124.

The 11th exercise will be the last one, and it will take place on Mon. Dec. 12th at 14.15.-16.00 in C124.

Information on the course

The lectures are on the weeks 36-42 and 44-50, Mondays 14-16 and Thursdays 14-16 in room C124, and the exercise group meets on Wednesdays 16-18 in C122, instructor Teemu Saksala. The course corresponds to 10 credit points, and it is on the advanced studies level. Prerequisites include B.Sc. level maths (aineopinnot) and in addition basics about complex numbers, Lebesgue integration and functional analysis; knowing some Sobolev space theory or at least the concept of the weak derivative would be useful.

We shall consider spectral theory of bounded and unbounded linear operators in Hilbert spaces. In this setting, spectrum is a generalization of the concept of eigenvalues of matrices. In addition to a treatment of the general theory, the goal of the course is to access the plentiful applications of spectral theory to partial differential equations and mathematical physics. Spectrum of the Schrödinger equation descibes the structure of atoms and molecules; spectrum of the Laplace-Dirichlet/Neumann problem in a bounded planar domain describes the vibrations of a drum membrane, or in a 3-d domain, propagation of electromagnetic waves; spectrum of the linearized elasticity system describes the mechanical vibrations in a solid and so on. The lectures on "Elliptic partial differential equations" (current schedule Autumn 2017) will be continuation of this course and will concentrate on the mentioned applications.

Topics include: unbounded operators in Hilbert spaces, symmetric and self-adjoint operators, self-adjoint extensions, definition of spectrum, eigenvalues, continuous, essential and residual spectrum, spectral theorem, spectrum of compact operators, max-min principle, applications to differential equations. The course material consists of lecture notes + additional reading.

Exams

There will be two examinations, one at the end of each period. The second course exam will be on Thursday Dec. 15th at 14.15-16.00 in C124. The maximum of each exam is 24 points, and to pass the course one has to get the minimum of 8 points in each exam. Bonus points from solutions of exercises: 25 % of problems solved = 1 point, 35 % = 2 points, 45 % = 3 points, 55 % = 4 points, 65 % = 5 points, 75 % = 6 points, to be added to the results of examinations.

Registration

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Exercises

Notice! You should solve the weekly exercise problems by Wed. afternoon! We go through the right solutions at the exercise session and can have some additional discussions on the topics of lectures.

Exercise problem sets as pdf-files can be downloaded from the links below.

Exercise 11