Dissipative Dirac operator with general boundary conditions on time scales

Abstract

UDC 517.9

In this paper, we consider the symmetric Dirac operator on bounded time scales. With general boundary conditions, we describe extensions (dissipative, accumulative, self-adjoint and the other) of such symmetric operators. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.

References

Lax, Peter D.; Phillips, Ralph S. Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London (1967).

Sz.-Nagy, Béla; Foiaş, Ciprian. Analyse harmonique des opérateurs de l'espace de Hilbert (French). Masson et Cie, Paris; Akadémiai Kiadó, Budapest (1967). https://doi.org/10.1002/zamm.19680480724

Allahverdiev, Bilender P. Spectral problems of nonselfadjoint 1D singular Hamiltonian systems. Taiwanese J. Math. 17, no. 5, 1487–1502 (2013). https://doi.org/10.11650/tjm.17.2013.2734

Allahverdiev, Bilender P. Extensions, dilations and functional models of Dirac operators. Integral Equations Operator Theory 51, no. 4, 459–475 (2005). https://doi.org/10.1007/s00020-003-1241-0

Allahverdiev, Bilender P. Spectral analysis of dissipative Dirac operators with general boundary conditions. J. Math. Anal. Appl. 283, no. 1, 287–303 (2003). https://doi.org/10.1016/s0022-247x(03)00293-2

Naĭmark, M. A. Линейные дифференциальные операторы (Russian), [Linear differential operators] Second edition, revised and augmented. With an appendix by V. È. Ljance, Nauka, Moscow (1969), 526 pp.

Gorbachuk, V. I.; Gorbachuk, M. L. Granichnye zadachi dlya differentsialʹno-operatornykh uravneniĭ (Russian), [Boundary value problems for operator-differential equations], Naukova Dumka, Kiev (1984), 284 pp. https://doi.org/10.1007/978-94-011-3714-0_2

Kuzhel, A. Characteristic functions and models of nonselfadjoint operators, Mathematics and its Applications 349, Kluwer Academic Publishers Group, Dordrecht (1996), x+273 pp. ISBN: 0-7923-3879-0

Pavlov, B. S. Self-adjoint dilation of a dissipative Schrödinger operator and eigenfunction expansion, Funct. Anal. and Appl. 98, 172–173 (1975).

Pavlov, B. S. Self-adjoint dilation of a dissipative Schrödinger operator and its resolution in terms of eigenfunctions, Math. USSR Sbornik 31, no. 4, 457–478 (1977).

Pavlov, B. S. Dilation theory and spectral analysis of nonselfadjoint differential operators (Russian), Mathematical programming and related questions (Proc. Seventh Winter School, Drogobych, 1974), Theory of operators in linear spaces, (Russian), pp. 3–69, Central. Èkonom. Mat. Inst. Akad. Nauk SSSR, Moscow (1976); English transl.: Transl. II. Ser., Amer. Math. Soc. 115, 103–142 (1981). https://doi.org/10.1090/trans2/115/06

Ginzburg, Yu. P.; Talyush, N. A. Exceptional sets of analytic matrix-functions, contracting and dissipative operators (Russian), Izv. Vyssh. Uchebn. Zaved. Mat., no. 8, 9–14 (1984), 82.

Ronkin, L. I. Введение в теорию целых функций многих переменных (Russian), [Introduction to the theory of entire functions of several variables], Nauka, Moscow (1971), 430 pp.

Weidmann, Joachim. Spectral theory of ordinary differential operators. Lecture Notes in Mathematics 1258, Springer-Verlag, Berlin (1987), vi+303 pp. ISBN: 3-540-17902-X. https://doi.org/10.1007/bfb0077960

Hilger, S. Ein Maßkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten (Ph. D. Thesis), Univ. Würzburg (1988).

Anderson, Douglas R.; Guseinov, Gusein Sh.; Hoffacker, Joan. Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math. 194, no. 2, 309–342 (2006). https://doi.org/10.1016/j.cam.2005.07.020

Atici, F. Merdivenci; Guseinov, G. Sh. On Green's functions and positive solutions for boundary value problems on time scales. Dynamic equations on time scales, J. Comput. Appl. Math. 141, no. 1-2, 75–99 (2002). https://doi.org/10.1016/S0377-0427(01)00437-X

Bohner, Martin; Peterson, Allan. Dynamic equations on time scales. An introduction with applications. Birkhäuser Boston, Inc., Boston, MA (2001), x+358 pp. ISBN: 0-8176-4225-0. https://link.springer.com/book/10.1007%2F978-1-4612-0201-1

Bohner, Martin; Peterson, Allan (Eds.), Advances in dynamic equations on time scales, Birkhäuser, Boston (2003).

Guseinov, Gusein Sh. Self-adjoint boundary value problems on time scales and symmetric Green's functions, Turkish J. Math. 29, no. 4, 365–380 (2005).

Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B. Dynamic systems on measure chains. Mathematics and its Applications 370. Kluwer Academic Publishers Group, Dordrecht (1996), x+285 pp. ISBN: 0-7923-4116-3. https://doi.org/10.1007/978-1-4757-2449-3

Levitan, B. M.; Sargsjan, I. S. Sturm–Liouville and Dirac operators, Translated from the Russian, Mathematics and its Applications (Soviet Series) 59, Kluwer Academic Publishers Group, Dordrecht (1991), xii+350 pp. ISBN: 0-7923-0992-8. https://doi.org/10.1007/978-94-011-3748-5

Thaller, Bernd. The Dirac equation, Texts and Monographs in Physics, Springer-Verlag, Berlin (1992), xviii+357 pp. ISBN: 3-540-54883-1

Rynne, Bryan P. $L^2$ spaces and boundary value problems on time-scales, J. Math. Anal. Appl. 328, no. 2, 1217–1236 (2007).

Gulsen, Tuba; Yilmaz, Emrah. Spectral theory of Dirac system on time scales, Appl. Anal. 96, no. 16, 2684–2694 (2017). https://doi.org/10.1080/00036811.2016.1236923

Guseinov, Gusein Sh. An expansion theorem for a Sturm–Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl. 3, no. 1, 147–160 (2008). http://campus.mst.edu/adsa/contents/v3n1p11.pdf

Guseinov, Gusein Sh. Eigenfunction expansions for a Sturm–Liouville problem on time scales, Int. J. Difference Equ. 2, no. 1, 93–104 (2007). http://campus.mst.edu/ijde/contents/v2n1p8.pdf

Huseynov, Adil; Bairamov, Elgiz. On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory 9, no. 1, 77–88 (2009). http://www.e-ndst.kiev.ua/v9n1/7(26)a.pdf

Allahverdiev, Bilender P.; Eryilmaz, Aytekin; Tuna, Hüseyin. Dissipative Sturm–Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electron. J. Differential Equations, Paper No. 95 (2017), 13 pp. https://ejde.math.txstate.edu/Volumes/2017/95/allahverdiev.pdf

Allakhverdiev, Bilender P. Extensions of symmetric singular second-order dynamic operators on time scales, Filomat 30, no. 6, 1475–1484 (2016). https://doi.org/10.2298/fil1606475a

Allahverdiev, Bilender P. Non-self-adjoint singular second-order dynamic operators on time scale, Math. Methods Appl. Sci. 42, no. 1, 229–236 (2019). https://doi.org/10.1002/mma.5338

Allahverdiev, Bilender P.; Tuna, Hüseyin. Spectral analysis of singular Sturm–Liouville operators on time scales, Ann. Univ. Mariae Curie-Skłodowska Sect. A 72, no. 1, 1–11 (2018). https://doi.org/10.17951/a.2018.72.1.1-11

Tuna, Hüseyin. Dissipative Sturm–Liouville operators on bounded time scales. Mathematica 56(79), no. 1, 80–92 (2014).

Tuna, Hüseyin. Completeness of the rootvectors of a dissipative Sturm–Liouville operators on time scales, Appl. Math. Comput. 228, 108–115 (2014). https://doi.org/10.1016/j.amc.2013.11.072

Tuna, Hüseyin. Completeness theorem for the dissipative Sturm–Liouville operator on bounded time scales, Indian J. Pure Appl. Math. 47, no. 3, 535–544 (2016). https://doi.org/10.1007/s13226-016-0196-1

Tuna, Hüseyin; Özek, Mehmet Afşin. The one-dimensional Schrödinger operator on bounded time scales, Math. Methods Appl. Sci. 40, no. 1, 78–83 (2017). https://doi.org/10.1002/mma.3966

Huseynov, Adil. Limit point and limit circle cases for dynamic equations on time scales. Hacet. J. Math. Stat. 39, no. 3, 379–392 (2010). http://www.hjms.hacettepe.edu.tr/uploads/8be9fcb8-9174-4966-a162-51221942b475.pdf

Özkan, A. S. Parameter dependent Dirac systems on time scales, Cumhuriyet Sci. J. 39, no. 4, 864–870 (2018). https://doi.org/10.17776/csj.471958

Published
29.04.2020
How to Cite
AllahverdievB. P., and TunaH. “Dissipative Dirac Operator With General Boundary Conditions on Time Scales”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 5, Apr. 2020, pp. 583–599, doi:10.37863/umzh.v72i5.546.
Section
Research articles