Quantum superintegrable systems with spin and physical applications

This project focuses on the systematic study of quantum superintegrable systems with spin, where additional internal degrees of freedom lead naturally to matrix-valued Hamiltonians and richer symmetry structures. A central aim is to understand how spin modifies classical notions of integrability and gives rise to new families of exactly or quasi-exactly solvable quantum models.

The research emphasizes the classification of superintegrable Hamiltonians admitting nontrivial integrals of motion, including first- and higher-order conserved operators. Particular attention is given to systems in the interaction of two non-relativistic particles with spin. We have recently classified all such superintegrable systems admitting additional first-order scalar integrals of motion. We are currently searching for systems with additional first-order pseudo-scalar and vector integrals of motion.

Beyond the classification problem, the project aims to identify physically meaningful models and explore their spectral properties, degeneracies, and potential applications in quantum mechanics and mathematical physics. The interplay between geometry, symmetry, and quantum dynamics is a unifying theme throughout the work.

Main topics include:

  • Classification of quantum superintegrable systems with spin
  • Integrals of motion and associated symmetry algebras
  • Exact solvability and spectral properties of spin-dependent models
  • Physical interpretations and applications of superintegrable systems

This research is supported by a TÜBİTAK 1001 Scientific and Technological Research Project and is conducted within the funding period from October 1, 2023 to April 1, 2026.

Project outputs: