| CODE | MAT5314 | ||||||||
| TITLE | Further Topics in Analysis | ||||||||
| UM LEVEL | 05 - Postgraduate Modular Diploma or Degree Course | ||||||||
| MQF LEVEL | 7 | ||||||||
| ECTS CREDITS | 15 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | - Fourier series: pointwise, uniform, Cesaro, and mean square convergence of Fourier series; the Weierstrass approximation theorem; - The Schrodinger equation. The harmonic oscillator and the hydrogen atom; - The Lebesgue theory of differentiation; absolutely continuous functions; the fundamental theorem of calculus; - Self-adjointness and spectral theory of bounded and unbounded operators in Hilbert space. Applications to Schrodinger operators and quantum mechanics. Study-unit Aims: The aim of this study-unit is to build on the analysis study-units of the undergraduate programme, and introduce students to applications of advanced analysis, particularly in Mathematical Physics. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Prove and apply the Fundamental Theorem of Calculus for Lebesgue Integration; - Prove various convergence theorems for Fourier series, and solve related problems; - Prove the spectral theorem for particular classes of linear operators in Hilbert Space. 2. Skills: By the end of the study-unit the student will be able to: - Identify absolutely continuous functions and solve problems related to these functions; - Prove convergence in the mean, or pointwise convergence, or Cesaro convergence of given Fourier series; - Apply the spectral theory of operators in Hilbert Space to problems in quantum mechanics. Main Text/s and any supplementary readings: - F. Jones, Lebesgue Integration on Euclidean Space, Jones and Bartlett (2000). - T. W. K枚rner, Fourier Analysis, Cambridge University Press (1989). - E. M Stein & R. Shakarchi, Fourier Analysis: An Introduction, Princeton University Press (2003). - E. M Stein & R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton University Press (2005). - E. Kreyszig, Introductory Functional Analysis with Applications, Wiley (2001). - S. J. Gustafson & I. M. Sigal, Mathematical Concepts of Quantum Mechanics, Springer (2003). - M. Reed & B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press (1981). |
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| ADDITIONAL NOTES | Pre-requisite Qualifications: B.Sc. with Mathematics as a main area Follows from: MAT3217 and MAT3211 |
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| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2025/6. It may be subject to change in subsequent years. |
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