Exploring Tridiagonal Realizations of Unitary Representations in su(1,1) Lie Algebra
The study explores tridiagonal (Jacobi-type) realizations of unitary representations of the Lie algebra su(1,1) in the space of square-summable sequences. It uncovers explicit similarity transformations that reduce these operators to diagonal form, noting that these transformations can be nonunitary and potentially unbounded. The research also highlights the nonuniqueness of diagonalization for representations of elliptic elements of the Lie algebra su(1,1), emphasizing the difference from spectral diagonalization. This variation leads to time-dependent diagonalizations for the corresponding one-parameter unitary groups.
The author acknowledges valuable discussions with L. Accardi, I.V. Volovich, O.V. Lychkovskiy, and V.Zh. Sakbaev on the topics addressed in the paper, as well as the reviewer for insightful comments that enriched the discussion of practical implications of the study. The research received support from the Russian Science Foundation under project no. 24-11-00039 at the Mathematical Institute, Russian Academy of Sciences, Moscow, Russia.
A. E. Teretenkov, the author of this study, affirms no conflicts of interest. The translation was conducted by I. Ruzanova. Pleiades Publishing maintains neutrality concerning jurisdictional claims in published maps and institutional affiliations. The article may have involved the use of AI tools for translation or editing purposes.
In conclusion, the research delves into the intricate diagonalization of unitary representations of the Lie algebra su(1,1), shedding light on the complexities and nuances of the process. The findings contribute to the understanding of the underlying mathematical structures and their implications in various applications.