Singular linear q-Hamiltonian systems

Bilender Allahverdiev, Huseyin Tuna

Abstract


In this paper, a singular linear \(q\)-Hamiltonian system is considered. The Titchmarsh-Weyl theory for this problem is constructed. Firstly, we provide some necessary fundamental concepts of the \(q\)-calculus. Later, we studied Titchmarsh-Weyl functions \(M\left(  \lambda\right)\) and circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) for this system. Circles \(\mathcal{C}_{TW}\left(a,\lambda\right)\) are proved to be nested. In the fourth part of the work, the number of square-integrable solutions of this system is studied. In the fifth  part of the work, boundary conditions in the singular case are obtained. Finally, a self-adjoint operator is defined.

Keywords


q-Hamiltonian system; singular point; Titchmarsh-Weyl theory.

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References


Allahverdiev, B. P., Tuna, H., q-Hamiltonian systems, Turkish J. Math. 44 (2020), 2241–2258.

Allahverdiev, B. P., Tuna, H., Singular discontinuous Hamiltonian systems, J. Appl. Analys. Comput. 12(4) (2022), 1386–1402.

Allahverdiev, B. P., Tuna, H., Singular Hahn–Hamiltonian systems, Ufa Math. J. 14(4) (2022), 3–16.

Annaby, M. H., Mansour, Z. S., q-Fractional Calculus and Equations, Springer, Heidelberg, 2012.

Atkinson, F. V., Discrete and Continuous Boundary Problems, Acad. Press Inc., New York, 1964.

Bangerezako, G., q-Difference linear control systems, J. Difference Equat. Appl. 17(9) (2011), 1229–1249.

Behncke, H., Hinton, D., Two singular point linear Hamiltonian systems with an interface condition, Math. Nachr. 283(3) (2010), 365–378.

Ernst, T., The History of q-Calculus and a New Method, U. U. D. M. Report (2000): 16, ISSN1101-3591, Department of Mathematics, Uppsala University, 2000.

Hinton, D. B., Shaw, J. K., On Titchmarsh–Weyl M(λ)-functions for linear Hamiltonian systems, J. Differential Equations 40(3) (1981), 316–342.

Hinton, D. B., Shaw, J. K., Titchmarsh–Weyl theory for Hamiltonian systems, in: Spectral theory of differential operators (Birmingham, Ala. 1981), 219–231, North-Holland, Amsterdam–New York, 1981.

Hinton, D. B., Shaw, J. K., Parameterization of the M(λ) function for a Hamiltonian system of limit circle type, Proc. Roy. Soc. Edinburgh Sect. A 93 (1983), 349–360.

Hinton, D. B., Shaw, J. K., Hamiltonian systems of limit point or limit circle type with both endpoints singular, J. Differ. Equat. 50 (1983), 444–464.

Kac, V., Cheung, P., Quantum Calculus, Springer-Verlag, New York, 2002.

Krall, A. M., Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhauser Verlag, Basel, 2002.

Krall, A. M., M(λ) theory for singular Hamiltonian systems with one singular point, SIAM J. Math. Anal. 20 (1989), 664–700.

Krall, A. M., M(λ) theory for singular Hamiltonian systems with two singular points, SIAM J. Math. Anal. 20 (1989), 701–715.

Yalcin, Y., Sumer, L. G., Kurtulan, S., Discrete-time modeling of Hamiltonian systems, Turkish J. Electric. Eng. Comput. Sci. 23 (2015), 149–170.




DOI: http://dx.doi.org/10.17951/a.2024.78.1.1-15
Date of publication: 2024-07-29 22:47:27
Date of submission: 2024-07-11 13:52:49


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