Boundary triples for integral systems

Strelnikov, DI
Dopov. Nac. akad. nauk Ukr. 2018, 7:3-9
Section: Mathematics
Language: Ukrainian

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
  1. Kochubei, A. N. (1975). On extensions of symmetric operators and symmetric binary relations. Math. Notes, 17, No. 1, pp. 25-28.
  2. 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:
  3. Gorbachuk, V. I. & Gorbachuk, M. L. (1984). Boundary problems for differential operator equations. Kiev: Naukova Dumka (in Russian).
  4. 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.
  5. 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:
  6. 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.
  7. 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:
  8. 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.
  9. 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).
  10. Atkinson, F. V. (1964). Discrete and continuous boundary problems. New York; London: Academic Press.
  11. 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).
  12. Bennewits, C. (1989). Spectral asymptotics for Sturm-Liouville equations. Proc. London Math. Soc., s3-59, Iss. 2, pp. 294-338. doi:
  13. 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.
  14. Arens, R. (1961). Operational calculus of linear relations. Pac. J. Math., 11, No. 1, pp. 9-23. doi:
  15. Strelnikov, D. (2017). Boundary triples for integral systems on finite intervals. Ukr. Math. Bull., 14, No. 3, pp. 418-439.