ECTS - Spectral Representations and Unbounded Operator Theory
Spectral Representations and Unbounded Operator Theory (MATH659) Course Detail
| Course Name | Course Code | Season | Lecture Hours | Application Hours | Lab Hours | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| Spectral Representations and Unbounded Operator Theory | MATH659 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Natural & Applied Sciences Master's Degree |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture, Discussion, Question and Answer. |
| Course Lecturer(s) |
|
| Course Objectives | This course is designed to give students an idea of the modern operator theory and its applications. After a short review of some classes of bounded linear operators on a Hilbert space, we will consider projection operators and the spectral family. Using the spectral family, spectral representation of self adjoint operators will be obtained. Then we will turn to the theory of unbounded linear operators. Spectral representations of unitary and consequently, not necessarily bounded self adjoint operators will be discussed. Finally, we will consider applications of unbounded operators in Quantum Mechanics and in particular the Heisenberg uncertainty principle. The course is aimed at Mathematics students who want to pursue a career in Analysis and its applications. |
| Course Learning Outcomes |
The students who succeeded in this course;
|
| Course Content | Projection operators and their properties on Hilbert Spaces, Spectral family, Spectral representation of self adjoint operators, Unbounded operators on a Hilbert space, Spectral representation of unitary operators and not necessarily bounded self adjoint operators, applications of unbounded operators in Quantum Mechanics. |
Weekly Subjects and Releated Preparation Studies
| Week | Subjects | Preparation |
|---|---|---|
| 1 | A Review of Hilbert space operators | [1], 3.10, 9.1, 9.2 |
| 2 | A Review of Hilbert space operators | [1], 9.3, 9.4 |
| 3 | Projection operators on a Hilbert space and the spectral family | [1], 9.5, 9.6, 9.7 |
| 4 | Spectral family of a bounded self adjoint operator | [1], 9.8 |
| 5 | Spectral representation of a bounded self adjoint operator | [1], 9.9, 9.10 |
| 6 | Spectral representation of a bounded self adjoint operator | [1], 9.9, 9.10 |
| 7 | Spectral properties of the spectral family of a bounded self adjoint operator | [1], 9.11 |
| 8 | Review and Midterm | |
| 9 | Hellinger – Toeplitz theorem Unbounded linear operators on a Hilbert space | [1], 10.1, 10.2 |
| 10 | Closed linear operators and closure | [1], 10.3 |
| 11 | Spectral properties of self adjoint operators Review of unitary operators | [1], 10.4 |
| 12 | Spectral representation of a unitary operator | [1], 10.5 |
| 13 | Cayley transform and the spectral representation of a self adjoint operator | [1], 10.6 |
| 14 | Operators of multiplication and differentiation | [1], 10.7 |
| 15 | States, observables, position and momentum operators, Heisenberg uncertainty principle | [1], 11.1, 11.2 |
| 16 | Review |
Sources
| Course Book | 1. E. Kreyszig, , Introductory Functional Analysis with Applications, Wiley Clas |
|---|---|
| 2. Yurij M. Berezansky, Zinovij G. Sheftel, Georgij F. Us, Functional Analysis Vol. II, Birkhӓuser,1996. |
Evaluation System
| Requirements | Number | Percentage of Grade |
|---|---|---|
| Attendance/Participation | - | - |
| Laboratory | - | - |
| Application | - | - |
| Field Work | - | - |
| Special Course Internship | - | - |
| Quizzes/Studio Critics | - | - |
| Homework Assignments | 3 | 30 |
| Presentation | 1 | 20 |
| Project | - | - |
| Report | - | - |
| Seminar | - | - |
| Midterms Exams/Midterms Jury | 1 | 20 |
| Final Exam/Final Jury | 1 | 30 |
| Toplam | 6 | 100 |
| Percentage of Semester Work | 70 |
|---|---|
| Percentage of Final Work | 30 |
| Total | 100 |
Course Category
| Core Courses | X |
|---|---|
| Major Area Courses | |
| Supportive Courses | |
| Media and Managment Skills Courses | |
| Transferable Skill Courses |
The Relation Between Course Learning Competencies and Program Qualifications
| # | Program Qualifications / Competencies | Level of Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 4 | Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary. | X | ||||
| 5 | Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 8 | Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 11 | Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 14 | 2 | 28 |
| Presentation/Seminar Prepration | 1 | 10 | 10 |
| Project | |||
| Report | |||
| Homework Assignments | 3 | 5 | 15 |
| Quizzes/Studio Critics | |||
| Prepration of Midterm Exams/Midterm Jury | 1 | 12 | 12 |
| Prepration of Final Exams/Final Jury | 1 | 12 | 12 |
| Total Workload | 125 | ||