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 | Ph.D. |
| 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 | Alanında, bağımsız olarak, bir problem kurgulayabilir, çözüm yöntemi geliştirerek problemi çözebilir ve sonuçları değerlendirebilir | X | ||||
| 2 | Matematiğin temel alanlarında ve kendi uzmanlığı olarak seçtiği alanda gerekli alt yapıyı oluşturur. | X | ||||
| 3 | Matematik literatürünü ve özel olarak kendi araştırma konusu ile ilgili ulusal ve uluslararası güncel yayınları takip edebilir ve bunlardan kendi araştırma konusu ile ilgili olanları çalışmalarında kullanabilir | X | ||||
| 4 | Bilimsel etik değerleri ve kuralları dikkate alır ve mesleki ve toplumsal yaşamda kullanabilir | X | ||||
| 5 | Kendi çalışmalarının sonuçlarını veya belli bir konudaki güncel çalışmaları ve bulguları, çeşitli bilimsel toplantılarda topluluk önünde Türkçe ve İngilizce olarak sunabilir ve tartışmalara katılabilir. | X | ||||
| 6 | Gerek bireysel, gerek bir çalışma grubunun üyesi olarak çalışabilme becerisini geliştirir | X | ||||
| 7 | Yaratıcı ve eleştirel düşünme, problem çözme, özgün bir çalışma üretme becerisini geliştirir. Bilimsel gelişmeleri takip eder, özümsediği bilgilerin analiz, sentez ve değerlendirmesini yapabilir. | X | ||||
| 8 | Kazandığı bilgi, beceri ve yetkinlikleri yaşam boyu geliştirmeye açık olur. | X | ||||
| 9 | Alanında özümsediği bilgiyi ve problem çözme yeteneğini disiplinler arası çalışmalarda uygulayabilir; karşılaşılan problemleri matematiksel modellerle ifade ederek, matematiksel bakış açısı ile farklı çözüm yöntemleri önerir. | X | ||||
| 10 | Matematik temelli yazılımları, bilişim ve iletişim teknolojilerini bilimsel amaçlı kullanabilir. | 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 | ||