Spectral Theory

Cod y Modiwl: MA38010
Teitl y Modiwl: Spectral Theory
Blwyddyn Academaidd: 2026/2027
Semester: Semester 2
Cyd-gysylltydd y Modiwl: Dr Robert Douglas
Rhestr Ddarllen: Gweld ar Aspire
Rhagofynion: MA30210 neu MT30210
Anghymharus (Unrhyw Flwyddyn Acad): MAM8020
Staff Eraill sy'n Cyfrannu: Dr Robert Douglas

Dulliau Asesu

Math o Asesiad	Manylion Asesiad	Hyd Asesiad	Cyfran
Arholiad Semester	Written Examination:	2 Awr	100%
Arholiad Ailsefyll	Written Examination:	2 Awr	100%

Canlyniadau Dysgu

Wedi cwblhau'r modiwl dylai'r myfyrwyr fedru:

Demonstrate knowledge of examples of Hilbert spaces and inner products.
Make use of orthogonality relations to manipulate inner products.
Expand elements of a Hilbert space in terms of an orthonormal basis.
Determine the norm of bounded linear operators.
Determine adjoints of operators and check for selfadjointness and compactness.
Define the spectrum and resolvent set.
Describe spectrum of compact and selfadjoint operators.

Disgrifiad cryno

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.

Nod

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.

Cynnwys

Introduction to Hilbert spaces: inner products, orthogonality, orthogonal complements, orthonormal systems, basis of a Hilbert space, examples
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

Sgiliau Modiwl

Math o Sgiliau	Manylion Sgiliau
Cyfathrebu	Written answers to exercises must be clear and well-structured.
Datblygu personol a chynllunio gyrfa	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.
Datrys Problemau	The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.
Gwaith Tim	Students will be encouraged to work on problems in groups during exercise classes.
Gwella dysgu a pherfformiad ei hun	Students are expected to develop their own approach to time-management regarding completion of assignments on time and preparation between lectures.
Rhifedd	Required throughout the course
Sgiliau pwnc penodol	Broadens exposure of student to topics in mathematics

Nodau

Mae'r modiwl hwn yn cydymffurfio a FfCChC Lefel 6