MATH 408 | Course Introduction and Application Information - Department of Mathematics

Course Name

Introduction to Spectral Analysis

Code	Semester	Theory (hour/week)	Application/Lab (hour/week)	Local Credits	ECTS
MATH 408	Fall/Spring	3	0	3	6

Prerequisites	None
Course Language	English
Course Type	Elective
Course Level	First Cycle
Mode of Delivery	-
Teaching Methods and Techniques of the Course	-
National Occupation Classification	-
Course Coordinator	-
Course Lecturer(s)	Dr. Öğr. Üyesi Sevin GÜMGÜM
Assistant(s)	Araş. Gör. Dr. Ayşe BELER Araş. Gör. Merve KAHRAMAN ARİMAN Araş. Gör. Sıla Selenay KOÇ

Course Objectives

The objective of this course is to teach the spectral theory of nonselfadjoint differential equations developed by Naimark, Levitan, Lyance and others.

Learning Outcomes

The students who succeeded in this course;

will be able to examine the structure and asymptotic behavior of Jost solutions of non-conjugate Strurm Liouville differential equations.
will be able to explain quantitative properties of the spectrum.
will be able to give spectral expansion in terms of associated vectors corresponding to eigenvalues and spectral singularities.
will be able to analyze the quantitative properties of spectrum using boundary theory.
will be able to solve using the Green function.

Course Description

This course aims to cover basic theory and applications of spectral analysis.

Course Category	Core Courses
	Major Area Courses
	Supportive Courses	X
	Media and Management Skills Courses
	Transferable Skill Courses

Week	Subjects	Related Preparation	Learning Outcome
1	Introduction and fundamental concepts	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
2	Fourier transforms; properties and applications	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
3	NonSelfadjoint differential equations	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
4	Non-Self adjoint Sturm Liouville differential operator	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
5	Solutions and their asymptotic behaviours	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
6	Jost solution and Its properties.	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
7	A special integral representation for Jost solution	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
8	Integral equations	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
9	The resolvent operator	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
10	Green’s function and its properties	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
11	Boundary uniqueness theorems of analytic functions	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
12	Beurling and Pavlov theorems, and their applications	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
13	Carleson’s theorem, and its applications	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
14	Quantitative properties of the spectrum, spectral expansion	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
15	Semester Review
16	Final Exam

Course Notes/Textbooks	"Linear Differential Operators: Two Volumes Bound as One" by M. A. Naimark, Dover Publications, 2014. ISBN-13: 978-0486468976
Suggested Readings/Materials	"Sturm Liouville and Dirac Operators" by B.M. Levitan, I. S. Sargsjan, Springer Netherlands, 1991. Hard Cover ISBN: 978-0-7923-0992-5

Semester Activities	Number	Weigthing
Participation
Laboratory / Application
Field Work
Quizzes / Studio Critiques
Portfolio
Homework / Assignments	5	20
Presentation / Jury
Project
Seminar / Workshop
Oral Exams
Midterm	1	40
Final Exam	1	40
Total

Weighting of Semester Activities on the Final Grade	6	60
Weighting of End-of-Semester Activities on the Final Grade	1	40
Total

Semester Activities	Number	Duration (Hours)	Workload
Theoretical Course Hours (Including exam week: 16 x total hours)	16	3	48
Laboratory / Application Hours (Including exam week: '.16.' x total hours)	16		0
Study Hours Out of Class	14	3	42
Field Work			0
Quizzes / Studio Critiques			0
Portfolio			0
Homework / Assignments	5	6	30
Presentation / Jury			0
Project			0
Seminar / Workshop			0
Oral Exam			0
Midterms	1	25	25
Final Exam	1	35	35
		Total	180

#	PC Sub	Program Competencies/Outcomes	* Contribution Level
#	PC Sub	Program Competencies/Outcomes	1	2	3	4	5
1	To be able to have a grasp of basic mathematics, applied mathematics or theories and applications of statistics.		-	-	-	X	-
2	To be able to use advanced theoretical and applied knowledge, interpret and evaluate data, define and analyze problems, develop solutions based on research and proofs by using acquired advanced knowledge and skills within the fields of mathematics or statistics.		-	-	X	-	-
3	To be able to apply mathematics or statistics in real life phenomena with interdisciplinary approach and discover their potentials.		-	-	X	-	-
4	To be able to evaluate the knowledge and skills acquired at an advanced level in the field with a critical approach and develop positive attitude towards lifelong learning.		-	-	-	X	-
5	To be able to share the ideas and solution proposals to problems on issues in the field with professionals, non-professionals.		-	-	-	-	-
6	To be able to take responsibility both as a team member or individual in order to solve unexpected complex problems faced within the implementations in the field, planning and managing activities towards the development of subordinates in the framework of a project.		-	-	-	-	-
7	To be able to use informatics and communication technologies with at least a minimum level of European Computer Driving License Advanced Level software knowledge.		-	-	-	-	-
8	To be able to act in accordance with social, scientific, cultural and ethical values on the stages of gathering, implementation and release of the results of data related to the field.		-	-	-	-	-
9	To be able to possess sufficient consciousness about the issues of universality of social rights, social justice, quality, cultural values and also environmental protection, worker's health and security.		-	-	-	-	-
10	To be able to connect concrete events and transfer solutions, collect data, analyze and interpret results using scientific methods and having a way of abstract thinking.		-	-	-	-	-
11	To be able to collect data in the areas of Mathematics or Statistics and communicate with colleagues in a foreign language.		-	-	-	-	-
12	To be able to speak a second foreign language at a medium level of fluency efficiently.		-	-	-	-	-
13	To be able to relate the knowledge accumulated throughout the human history to their field of expertise.		-	-	-	-	-