Module Information
Course Delivery
Assessment
| Assessment Type | Assessment length / details | Proportion |
|---|---|---|
| Semester Exam | 2 Hours Examination | 100% |
| Supplementary Exam | 2 Hours Examination | 100% |
Learning Outcomes
On successful completion of this module students should be able to:
1. demonstrate knowledge of examples of Hilbert spaces and inner products
2. make use of orthogonality relations to manipulate inner products
3. expand elements of a Hilbert space in terms of an orthonormal basis
4. determine the norm of bounded linear operators
5. determine adjoints of operators and check for selfadjointness and compactness
6. define the spectrum and resolvent set
7. describe spectrum of compact and selfadjoint operators
Aims
Spectral Theory is one of the main current areas of research in mathematical analysis with applications in many sciences, in particular quantum mechanics. It underpins much of the modern theory of solutions of partial differential equations.
Brief description
Spectral theory deals with solvability of equations of the form (T-z)x=y, where T is a linear, but not necessarily bounded operator on a Banach or Hilbert space and z is a complex number. This module aims to introduce the basic concepts needed from Hilbert space theory and the theory of linear operators and give some first results on the spectrum of operators.
Content
Linear operators on Hilbert spaces: bounded operators, adjoint operators, selfadjoint operators, compact operators, examples
Spectral theory: spectrum, resolvent, Neumann series, spectral radius for bounded operators, spectrum of compact operators, spectrum of selfadjoint operators, the spectral theorem for compact selfadjoint operators, examples
Module Skills
| Skills Type | Skills details |
|---|---|
| Application of Number | Required throughout the course |
| Communication | Written answers to exercises must be clear and well-structured. |
| Improving own Learning and Performance | Students are expected to develop their own approach to time-management regarding completion of assignments on time and preparation between lectures. |
| Information Technology | N/A |
| Personal Development and Career planning | Completion of tasks (assignments) to set deadlines will aid personal development. The course will give indications of whether a student wants to further pursue mathematical analysis and its applications. |
| Problem solving | The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills. |
| Research skills | N/A |
| Subject Specific Skills | Broadens exposure of student to topics in mathematics |
| Team work | Students will be encouraged to work on problems in groups during exercise classes. |
Notes
This module is at CQFW Level 6