Penn State Acoustics

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 (see for example, 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.
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.

Exercise problems can be found via the link below. Solutions can be found under the course material link.

The second exam will take place Thursday May 3rd at 14.15-16.00 in the room C122. The material for this exam is the rest of the lecture note (starting from Section 6.4.) and exercises 6-10.

11.12.2017 klo 09:00 - 2.5.2018 klo 23:59


Käyttäjän Jari J Taskinen kuva

Jari J Taskinen

Julkaistu, 23.1.2018 klo 8:19

Exercise class will take place in CK107 on Wednesdays at 14.30-16. It is an instruction class with JT present. The solutions must be returned to JT on the next Friday at latest.


Tästä osiosta löydät kurssin opetusaikataulun. Tarkista mahdolliset muut aikataulut kuvauksesta.

Ti 16.1.2018
10:15 - 12:00
Ke 17.1.2018
12:15 - 14:00
Ti 23.1.2018
10:15 - 12:00
Ke 24.1.2018
12:15 - 14:00
Ti 30.1.2018
10:15 - 12:00
Ke 31.1.2018
12:15 - 14:00
Ti 6.2.2018
10:15 - 12:00
Ke 7.2.2018
12:15 - 14:00
Ti 13.2.2018
10:15 - 12:00
Ke 14.2.2018
12:15 - 14:00
Ti 20.2.2018
10:15 - 12:00
Ke 21.2.2018
12:15 - 14:00
Ti 27.2.2018
10:15 - 12:00
Ke 28.2.2018
12:15 - 14:00
Ti 13.3.2018
10:15 - 12:00
Ke 14.3.2018
12:15 - 14:00
Ti 20.3.2018
10:15 - 12:00

Muu opetus

24.01. - 28.02.2018 Ke 14.15-16.00
14.03.2018 Ke 14.15-16.00
Jari Taskinen
Opetuskieli: englanti



Kurssin suorittaminen

There will be two examinations, one at the end of each period. 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, in addition to the usual minimum of about 22 points total. 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.

The second exam will take place Thursday May 3rd at 14.15-16.00 in the room C122.


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