Strelnikov, DI |

Dopov. Nac. akad. nauk Ukr. 2018, 7:3-9 |

https://doi.org/10.15407/dopovidi2018.07.003 |

Section: Mathematics |

Language: Ukrainian |

Abstract: An integral system that contains the Sturm—Liouville equation, Stieltjes string, and Krein—Feller string as special cases is considered. The maximal and minimal linear relations associated with the system are studied in a connected Hilbert space. Boundary triples and corresponding Weyl functions for the maximal linear relation are constructed in both limit circle and limit point cases. |

Keywords: boundary triple, deficiency indices, integral system, symmetric linear relation, Weyl function |

References:

- Kochubei, A. N. (1975). On extensions of symmetric operators and symmetric binary relations. Math. Notes, 17, No. 1, pp. 25-28. doi: https://doi.org/10.1007/BF01093837
- Malamud, M. M. (1992). On the formula of generalized resolvents of a nondensely defined Hermitian operator. Ukr. Math. J., 44, Iss. 12, pp. 1522-1547. doi: https://doi.org/10.1007/BF01061278
- Gorbachuk, V. I. & Gorbachuk, M. L. (1984). Boundary problems for differential operator equations. Kiev: Naukova Dumka (in Russian).
- Derkach, V. A. & Malamud, M. M. (2017). Extension theory of symmetric operators and boundary value problems. Proceedings of Institute of Mathematics NAS of Ukraine, Vol. 104 (p. 573). Kyiv: Institute of Mathematics of the NAS of Ukraine.
- Lesh, M. & Malamud, M. (2003). On the deficiency indices and self-adjointness of symmetric Hamiltonian systems. J. Diff. Equat., 189, No. 2, pp. 556-615. doi: https://doi.org/10.1016/S0022-0396(02)00099-2
- Mogilevskii, V. (2009). Boundary triplets and Titchmarsh—Weyl functions of differential operators with arbitrary deficiency indices. Methods Func. Anal. Topol., 15, No. 3, pp. 280-300.
- Behrndt, J., Hassi, S., de Snoo, H. & Wietsma, R. (2011). Square-integrable solutions and Weyl functions for singular canonical systems. Math. Nachr., 284, No. 11-12, pp. 1334-1384. doi: https://doi.org/10.1002/mana.201000017
- Mogilevskii, V. (2015). Spectral and pseudospectral functions of Hamiltonian systems: development of the results by Arov-Dym and Sakhnovich. Methods Funct. Anal. Topol., 21, No. 4, pp. 70-402.
- Kac, I. S. (2002). Linear relations generated by a canonical differential equation of phase dimension 2 and decomposability in eigenfunctions. Algebra i Analiz, 14, No. 3, pp. 86-120 (in Russian).
- Atkinson, F. V. (1964). Discrete and continuous boundary problems. New York; London: Academic Press.
- Kac, I. S. & Krein, M. G. (1968). On the spectral functions of the string. Supplement 2 to the Russian translation of F.V. Atkinson. Discrete and continuous boundary problems (pp. 648-737). Moscow: Mir (in Russian).
- Bennewits, C. (1989). Spectral asymptotics for Sturm-Liouville equations. Proc. London Math. Soc., s3-59, Iss. 2, pp. 294-338. doi: https://doi.org/10.1112/plms/s3-59.2.294
- Arov, D. Z. & Dym, H. (2012). Bitangential direct and inverse problems for systems of integral and differential equations. Encyclopedia of Mathematics and its Applications, Vol. 145. Cambridge: Cambridge Univ. Press. doi: https://doi.org/10.1017/CBO9781139093514
- Arens, R. (1961). Operational calculus of linear relations. Pac. J. Math., 11, No. 1, pp. 9-23. doi: https://doi.org/10.2140/pjm.1961.11.9
- Strelnikov, D. (2017). Boundary triples for integral systems on finite intervals. Ukr. Math. Bull., 14, No. 3, pp. 418-439.