Kuantum Süperintegrallenebilir Sistemler ve Fiziksel Uygulamaları
Tuncer O. O., Kürkcüoğlu S., Yurduşen I. (Yürütücü)
TÜBİTAK Projesi, 1001 - Bilimsel ve Teknolojik Araştırma Projelerini Destekleme Programı, 2023 - 2026
- Proje Türü: TÜBİTAK Projesi
- Destek Programı: 1001 - Bilimsel ve Teknolojik Araştırma Projelerini Destekleme Programı
- Başlama Tarihi: Ekim 2023
- Bitiş Tarihi: Nisan 2026
Proje Özeti
In Mathematical Physics, superintegrable systems (SIS) are very important as they can be solved analytically
without numerical approaches or perturbative methods. Thus, they can be used to study the behavior of physical
systems and test new theoretical ideas. Mathematically, they make it easier to reach solutions with their addi-
tional symmetry properties. The purpose of this project is to study physically important quantum SIS, produce
pioneering publications, and investigate their applications. Our motivation is to study and classify systemati-
cally SIS involving two non-relativistic quantum particles with spin s = 1/2. Specifically, the following research
problems will be mainly discussed:
1. Finding and classifying SIS with two-spin for first/second-order integrals of motion (IOM),
2. Generating symmetry algebras of the resulting SIS,
3. Studying and classifying SIS with two-spin and higher-order IOM,
4. Obtaining analytical solutions, investigating the physical results and applications arising from these solu-
tions.
These problems will be the first studies in terms of systematic classification of SIS with two-spin. The systematic
investigation of SIS with one-spin was initiated by Winternitz and Yurdu¸sen (2006), and by (Winternitz ve Yur-
du¸sen, 2009; Désilets vd., 2012; Yurdu¸sen vd., 2021), the classification was completed for second-order IOM.
Previously, analytical solution for one non-trivial superintegrable system with spin and the corresponding phys-
ical application was given by Winternitz and Yurdu¸sen (2009). Therefore, the reason we choose quantum SIS
is their importance in the literature, their popularity, and that the questions to be solved will draw the attention of
experts. In addition, this project has original aspects because it focuses on quantum SIS that have not been
studied systematically, aims to classify SIS with two-spin, and plans to fill the gap in the literature by giving new
physical applications with analytical solutions.
In the specified classification problem, the walkthrough and the method to use are roughly as follows:
• Finding all scalar, vector, and tensor operators,
• Obtaining the most general operator by taking linear combinations of these operators with arbitrary weight
functions and symmetrizing this operator using the Mathematica code in (Yurdu¸sen and Tuncer, 2022),
• Finding the commutation of the symmetrized operator with the Hamiltonian, solving the determining equa-
tions obtained from this commutativity condition, and finding IOM,
• Classifying and interpreting the resulting SIS.
To generate the symmetry algebras, the commutations of the IOM will be expressed in terms of the
other integrals in the system. For this, a Mathematica package, NCAlgebra (Non-commutative algebra,
www.github.com/NCAlgebra/NC), will be used. To find new higher-order IOM, mainly the method of classify-
ing IOM up to the second-order will be followed, but the huge number of determining equations will make it
challenging to obtain solutions. Thus, we will first focus on third/fourth-order IOM, and inspired by the paper
(Escobar-Ruiz vd., 2018a) we will try to group the most general operator given at the beginning of the problem.
To investigate the physical applications of the resulting SIS, first, analytical solutions for these systems will be
found. Their existence will be checked in the literature, and for both cases, the theoretical and technological
benefits of analytical solutions for physical models will be examined.
This project will be conducted by a coordinator and two researchers (Prof. Seçkin Kürkcüoglu ve
Dr. Orhan Ogulcan Tuncer). They have recent studies on the subject and both researchers will take part in the
entire project. Besides, 2 undergraduate, 2 master, and 2 doctoral scholars will take on various tasks such as
literature review and solving equations through the Mathematica program.
Based on the specified problems, as the widespread impact we aim
• at least three publications in prestigious journals in SCI-Expanded,
• at least three presentations at national/international conferences,
• a workshop attended by researchers and students.
Additionally, we plan to contribute to academic education by enabling graduate scholars to write their dissertations in this subject.