Elgiz BAIRAMOV, Ekin UĞURLU

Fourth order differential operators with distributional potentials

In this paper, regular and singular fourth order differential operators with distributional potentials areinvestigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved,deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators areconsidered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such thata squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipativeand maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given inthe single and direct sum Hilbert spaces.

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[1] Bennewitz C, Everitt WN. On second-order left-definite boundary value problems, In: Everitt WN, Lewis RT (editors) Ordinary Differential Equations and Operators 1982. Berlin, Germany: Springer, 1983.
[2] Dunford N, Schwartz JT. Linear Operators, Part II, Spectral Theory. New York, USA: Interscience Publishers, 1964.
[3] Eckhardt J, Gesztesy F, Nichols R, Teschl G. Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials. Opuscula Mathematica 2013; 33 (3): 467-563.
[4] Everitt WN. Fourth order singular differential equations. Mathematische Annalen 1963; 149: 320-340.
[5] Everitt WN, Markus L. Boundary Value Problems and Sympletic Algebra for Ordinary Differential and QuasiDifferential Operators. Mathematical Surveys and Monographs 61. Rhode Island, USA: American Mathematical Society, 1999.
[6] Fulton CT. The Bessel-squared equation in the lim-2, lim-3, and lim-4 cases. The Quarterly Journal of Mathematics (2) 1989; 40: 423-456.
[7] Gorbachuk VI, Gorbachuk ML. Boundary Value Problems for Operator Differential Equations. Dordrecht, Netherlands: Springer, 1991.
[8] Bairamov E, Öztürk MR, Ismailov Z. Selfadjoint extensions of a singular differential operator. Journal of Mathematical Chemistry 2012; 50: 1100-1110.
[9] Ismailov Z, Öztürk MR. Normal extensions of a singular multipoint differential operator of first order. Electronic Journal of Differential Equations 2012; 36: 9.
[10] Ismailov Z, Erol M. Normal differential operators of first-order with smooth coefficients. Rocky Mountain Journal of Mathematics 2012; 42: 633-642.
[11] Ismailov Z. Selfadjoint extensions of multipoint singular differential operators. Electronic Journal of Differential Equations 2013; 231: 1-10.
[12] Ismailov Z, Sertbaş M, Güler B. Normal extensions of a first order differential operator. Filomat 2014; 28: 917-923.
[13] Ismailov Z, Guler OB, Ipek P. Solvability of first order functional differential operators. Journal of Mathematical Chemistry 2015; 53: 2065-2077.
[14] Ismailov Z, Ipek P. Selfadjoint singular differential operators of first order and their spectrum. Electronic Journal of Differential Equations 2016; 21: 1-9.
[15] Kodaira K. On ordinary differential equations of any even order and the corresponding eigenfunction expansions. American Journal of Mathematics 1950; 72: 502-544.
[16] Maozhu Z, Sun J, Zettl A. The spectrum of singular Sturm-Liouville problems with eigenparameter dependent boundary conditions and its approximation. Results in Mathematics 2013; 63: 1311-1330.
[17] Naimark MA. Linear Differential Operators. Ungar, New York, USA: George G. Harrap and Company, 1968.
[18] Savchuk AM, Shkalikov AA. Sturm-Liouville operators with singular potentials. Mathematical Notes 1999; 66 (6): 741-753.
[19] Weidmann J. Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, Berlin, Germany: Springer, 1987.
[20] Weyl H. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen wilkürlichen Funktionen. Mathematische Annalen 1910; 68: 220-269 (in German).