INTEGRAL REPRESENTATION FOR THE STURM-LIOUVILLE OPERATOR WITH DISCONTINUOUS COEFFICIENT

In this paper,the Sturm-Liouville with discontinuous coe¢ cientis considered. ·Integral equations for solution which satisfy certain initial ofgiven and equation has been obtained and useful integral representation forSturm-Liouville operator.

Keywords:

Integral equation Sturm-Liouville, kernel,

PDF

___

V.A. Ambartsumyan, Über eine Frage der Eigenwerttheorie, Z. Physik 53 (1929), 690-695.
G.Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimnungder Dif- ferentialgleichung durch die Eigenwerte, Acta Math. 78 (1945), 1-96.
N. Levinson, 1949, The Inverse Sturm-Liouville Problem, Mat. Tidsskr. B., pp. 25-30.
B. M. Levitan, 1964, Generalized Translation Operators and some of its Applications, Jerusalem.
V.A. Marchenko, Some Problems in the Theory of Second-order DiferentialOperators, Dokl. Akad., Nauk SSSR. 72 (1950), 457-560.
I. M. Gelfand and B. M. Levitan, 1951 On Determination of a Di¤erentialEquation by its
Spectral Function Izv. Akad. Nauk SSSR Ser. Mat. 15 309-60 (in Russian). M.G. Gasimov and B. M. Levitan, About Sturm-Liouville Di¤erential. Operators., Math. Sborn., 63 (105), No. 3, (1964).
A.N. Tikhonov, Uniqueness Theorems for Jeophysics Problems, Dokl. Akad. Nauk. SSSR, Vol 69, No 4, 1949, 797-800.
R. O. Hryniv and Ya. V. Mykytyuk, Inverse Spectral Problems for Sturm- Liouville Operators with Singular Potentials, Inverse Problems 19 (2003) 665-684.
A. M. Savchuk and A. A. Shkalikov, 1999 Sturm-Liouville Operators with Singular Potentials
Mat. Zametki 66 897-912 (in Russian). V. A. Marchenko 1950 Some Questions of the Theory of Second Order Dixoerential Operators
Dokl. Akad. Nauk SSSR 72 457-60. O. Hald, 1984 Discontinuous Inverse Eigenvalue Problem Commun. Pure Appl. Math. 37 539
R. Carlson, 1994 Inverse Sturm-Liouville Problems with a Singularity at Zero Inverse Prob- lems 10 851-64.
V. A. Yurko, 2000 Inverse Problems for Di¤erential Equations with Singularities Lying Inside the Interval J. Inverse III-Posed Probl. 8 89-103.
R. Kh. Amirov and V. A. Yurko, On Di¤erential Operators with Singularity and Discontinuity Conditions Inside the Interval. Ukr. Math. Jour., 2001, v.53, No11, p. 1443-1458.
R. Kh. Amirov, Transformation Operators for Strum-Liouville Operators with Singular Po- tentials, Int. Journal of Pure and Applied Math., 12(3), (2004).
R. Kh Amirov, Direct and Inverse Problems for Di¤erential Operators with Singularity and Discontinuity Conditions Inside the Interval Transactions of NASAzerbaijan. R. Kh Amirov, On Sturm-Liouville Operators with Discontinuity Conditions Inside an Inter- val J. Math. Anal. Appl. 317 (2006) 163-176.
Nilifer Topsakal, Sonlu Aralıkta Coulomb Potansiyele Sahip Sturm- Liouville Operatörü Için Ters ( Inverse) Problemler (2008)
S. Albeverio and P. Kurasov, 2000 Singular Perturbations of Di¤erential Operators, Solvable
Schrödinger Type Operators (Cambridge: Cambridge University Press) J. Pöschel and E. Trubowitz, 1987 Inverse Spectral Theory(Pure Appl. Math. Vol 130)(Or- lando, FL:Academic).