Multipoint selfadjoint quasi-differential operators for first order

Year 2019, Volume: 68 Issue: 1, 964 - 972, 01.02.2019

Rukiye Öztürk Mert Bülent Yılmaz , Zameddin İ. Ismailov

https://doi.org/10.31801/cfsuasmas.501414

Abstract

In the present paper, the aim is to described all selfadjoint extensions of the minimal operator generated by first order linear symmetric multipoint quasi-differential operator expression in the direct sum of weighted Hilbert spaces of vector-functions defined at the semi-infinite intervals by using the Calkin-Gorbachuk method. We have also examine the structure of the spectrum of such extensions.

Keywords

Symmetric and selfadjoint differential operators , multipoint quasi-differential expression , deficiency indices , spectrum

References

• Bairamov E., Öztürk Mert, R, Ismailov, Z., Selfadjoint Extensions of a Singular Differential Operator, Journal of Mathematical Chemistry, 50: (2012), 1100-1110.
• El-Gebeily, M.A., O'Regan, D., Agarwal, R., Characterization of Self-adjoint Ordinary Differential Operators, Mathematical and Computer Modelling, 54 (2011), 659-672.
• Everitt, WN, Markus L., The Glazman-Krein-Naimark Theorem for Ordinary Differential Operators, Operator Theory, Advances and Applications, 98 (1997), 118-130.
• Everitt, W.N., Poulkou A., Some Observations and Remarks on Differential Operators Generated by First-Order Boundary Value Problems, Journal of Computational and Applied Mathematics, 153 (2003), 201-211.
• Glazman, IM., On the Theory of Singular Differential Operators, Uspehi Math. Nauk., 40 (1950), 102-135 (English translation in Amer. Math. Soc. Translations 1962; (1), 4: 331-372).
• Gorbachuk, VI, Gorbachuk, ML., Boundary Value Problems for Operator-Differential Equations, First ed., Kluwer Academic Publisher: Dordrecht, 1991.
• Hörmander, L., On the Theory of General Partial Differential Operators, Acta Mathematica, 94 (1955), 161-248.
• Naimark, MA., Linear Differential Operators II. Ungar, New York, 1968.
• von Neumann, J., Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1929-1930), 49-131.
• Rofe-Beketov, FS, Kholkin, AM., Spectral Analysis of Differential Operators, World Scientific Monograph Series in Mathematics v.7. 2005.
• Stone, MH., Linear Transformations in Hilbert Spaces and Their Applications in Analysis, American Math. Soc. Coloq., (1932), 15.
• Zettl, A, Sun, J., Survey Article: Self-adjoint Ordinary Differential Operators and Their Spectrum, Rocky Mountain Journal of Mathematics, 45, 1 (2015), 763-886.