Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi

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1 Journal of Engineering atural Sciences Mühendisli ve Fen Bilileri Dergisi Siga 00/ CALCULATED OF REGULARIZED TRACE OF A EVE ORDER DIFFERETIAL EQUATIO I FIITE ITERVAL İnci ALBAYRAK *, Fata AKGÜ Yıldız Teni Üniversitesi, Kiya-Metalurji Faültesi, Mateati Mühendisliği Bölüü, Davutpaşa-İSTABUL Geliş/Received: Kabul/Accepted: ABSTRACT In this study,the regularized trace forula of the even ordered differential equation with two ters given on a finite interval is exained Keywords: Eigenvalue, eigenfunction, asyptotic forula, unitar atrix. SOLU ARALIKTA ÇİFT MERTEBEDE BİR DİFERASİYEL DEKLEMİİ DÜZELİ İZİİ HESAPLAMASI ÖZET Bu çalışada sonlu aralıta veriliş çift ertebeden ii terili bir diferansiyel denlein düzenli iz forülü inceleniştir. Anahtar Sözcüler: Özdeğer, özfonsiyon, asitoti ifade, unitar atris.. ITRODUCTIO As nown, trace of n diensional atrix is the su of n eigenvalues. The eigenvalues of Stur- Liouville proble satisfy the following condition λ < λ <... < λ... n < li λ n n λ Fro here, it is seen that it hasn t finite liit of su of nubers { } because of this there is no eaning to say about the trace of the Stur-Liouville proble directly. Firstly; I.M. Gelf B.M. Levitan found the series su which was fored by nubers { } λ. The su of the series was called regularized trace of the Stur-Liouville proble. Regularized trace forula which was entioned was found with ore siple ethod by * Sorulu Yazar/Corresponding Autor: e-ail: ibayra@yildiz.edu.tr, Tel: (0)

2 Calculated of Regularized Trace of a Even Order L.A. Diii [] Later, the articles [5] - [] the boos [] - [] were written about this subject. We calculated the regularized trace of an even order differential equation with Diii s ethod. In this paper, the regularized trace forula for the operator L in L (0, ) space which is defined with differential equation ( ) ( ) y q( x) y λ y, 0 < x < () with boundary conditions " ( ) " ( ) y(0) y (0)... y (0) y( ) y ( )... y ( ) 0 () is calculated Here, qx ( ) is a real valued, continuous function in [0, ]. Initially, the regularized trace forula of Stur-Liouville operator was calculated by I.M.Gelf B.M.Levitan []. Later, this study was generalized for different operators. Excessive inforations references that are given in [],[] [5]-[],can be shown as recent studies. Trace forula obtained in [], was proved by an algebric ethod by L.A.Diii []. We obtained the regularized trace forula of ()-() by Diii ethod which was found before in []. While qx ( ) 0,it is clear that µ n ( n,,...) n are the eigenvalues of operator L ψ n ( x) sinnx are the orthonoral eigenfunctions corresponding to this eigenvalues. As nown fro [](see also[3]) the eigenvalues the orthonoral eigenfunctions of L have following asyptotic expressions. λ n ( ) n q x dx O 0 n ϕ( x) ψ ( x) O n n The results which were obtained are given at following theore. Theore: If qxdx ( ) 0, then 0 q(0) q( ) ( λ ) The series, at the left side of this equality are called reqularized trace of operator L. Proof: To prove this theore, it is sufficient to show that [ ϕ L ϕ ψ L ψ ] (3) n n n n li (, ) (, ) 0 n Really, by considering the following equations ϕ, Lϕ λ, ( ψ, Lψ ) n ( ψ, qψ ). ( ) n n n n n n n 73

3 İ. Albayra, F. Agün Siga 00/ It can be written as [( ) ( )] ( ) ψ, Lψ ϕ, Lϕ n λ ( ψ, qψ ) n n n n n. n n n n n Let we write ( ψ, qψ ) suation as below n n n ( ψ, qψ ) q( x)sin nxdx n n n n 0 cos(.0) q( x) cos xdx 0 n n cos. q( x) cos xdx 0 Here, if the expansion forula of ( ) q ( ψ qψ ) n n li, n is seen. (0) q( ) qx is considered according to cosinus in [ ] 0,, Considering the study [],we survey the transfer atrix fro orthonoral basis { ϕ i } to orthonoral basis { ψ j }. U is a Unitar atrix. Fro the forula i U i U i i Lψ ψ q( x) ψ it can be written as Here, if we consider that ( Lψ ϕ ) ( ψ ϕ ) ( qψ ϕ ),,,. i i i ( Lψ, ϕ ) ( ψ, Lϕ ) λ ( ψ, ϕ ) i i i i ( ), (, ) ( q, ) λ ψ ϕ ψ ϕ ψ ϕ i i i i () ( λi )( ψ, ϕ ) i ( qψ, ϕ ) i, fro the last expression it is found that i ( λ ) i ( ) ( ) i i λ ψ ϕ ψ, q < i n U < (5) i 7

4 Calculated of Regularized Trace of a Even Order Because of 0 q( x) dx 0,the asyptotic forulas that we entioned before, obtained as λ O( ), ϕ ψ O( ) Fro the equality (5), by doing siilar operations for every natural nuber while, it is seen that ( λ λ ) U < (6) i i i ( ) is true. According to [],if we consider it becoes that ( ψ, Lψ ) λu i i i ( ϕ, Lϕ ) λu i i [ ( ψ, Lψ ) ( ϕ, Lϕ )] ( ) U λ λ λ i i ( U U i i) (7) i i According to (6) ( λ λ ) U < i i i While, the right side of above inequality is written as dx n ( ) ( ) ( ) x x d (( ) )(( ) ) ( ) x ( ) ( ) ( ) du u u u du ( ) du u du 75

5 İ. Albayra, F. Agün Siga 00/ du ( ) u ( ) ( ) ( ) u du du (8) u ( ) for siplicity, by getting, we restrict the last su as: du du du u u u ( ) ( ) du du ( ) u ( ). n ( ) (9) ( ) Thus while, fro the (8) (9), while, ( λ λ ) u n 0 i i i ( ) is found. Thus, while, it is showed that the first su of the right side of (7) approachs to zero.the second su in (7) is restricted as below. If the below equation u u ( ϕ ψ, ϕ ψ ) i i i i the asyptotic expression of eigenfunctions of L is considered u u ϕ ψ. ϕ ψ i i i i i. is obtained. 76

6 Calculated of Regularized Trace of a Even Order If the siilar operations are done according to [], u u < i i ( ) is found. Here, fro the second su of (7) λ u u < i ( ) i i < ( ) ( ) ( ) ( ) n 0 is obtained. Thus, it satisfies the equality (3) according this accuracy, the theore is proved. As an acnowledgeent we want to express our gratitude to M.Bayraoğlu for his support to this wor. REFERECES [] Gelf, I. M. Levitan B.M. On a siple identity for the eigenvalues of a differential operatör if the second order Dol. Aad. au SSSR, M.T. 88, o:, (953). [] Sadovnichi V.A., Trace forulas of the higher order differential operators, Mate.sb. T.7, vıp., (967). [3] Levitan, B.M. Sargasyan, I.S., Stur-liouville Dirac Operators, Kluwer, Dordrecht, (99). [] Sadovnichi V.A. Theory of Operators, Consultants Bureau, ew-yor, 99. [5] Fulton, C.T. Pruesses, S. Eigenvalue eigenfunctions asyptotics for reguler Stur-Liouville probles, J. Math. Anal. Appl. 88, (99). [6] Bayraoğlu, M. Adıgüzelov, E.E. On a regularized trace forula for the Sturliouville operator with a bounded operator coefficient with a singularity. Differential Equations, no., (997). [7] Albayra I., Bayraoğlu M. Adıgüzelov, E.E. The second regularized trace forula for the Stur-Liouville proble with spectral paraeter in a boundary condition. Methods of Functional Analysis Topology V.6, o:3, -8, (000). [8] Albayra, I., Bayal, O. Gul E., Forula for the Highly Regularized Trace of the Stur-Liouville Operator with Unbounded Operator Coefficients Having Singularity, Tur J Math 5 (00), [9] Adıgüzelov, E.E., Avcı, H. Gul E. The trace forula for Stur-Liouvile operator with operator coefficient. J. Math. Phys. Vol., 6-6 o:6, (00). [0] Adıgüzelov, E.E. Bayal O., Bayraov A., On the spectru regularized trace of the Stur-Liouville proble with spectral paraeter on the boundary condition with the operator coefficient, International Journal of differential equations applications, V:, no:3, page (00). 77

7 İ. Albayra, F. Agün Siga 00/ [] Adıgüzelov, E.E. Başi Ö. On the regularized trace of the differential operator equation in a finite interval, Siga Journal of Engineering atural Sciences Issue (00). [] Diii, L.A. About a forula of Gelt-Levitan, Usp. Math. au, 8(), 9-3 (953). [3] aiar, M.A. Linear differential operators, part I. Harrap, London (968). 78