The regularized trace formula of the spectrum of a Sturm- Liouville problem with turning point

Zaki F.A. El-Raheem, Shimaa A.M. Hagag

Abstract


In this work, we establish the summation formula of the infinite sequence of the eigenvalues generated by a Sturm-Liouville problem with turning points in a finite interval [0, π].

Keywords


Singular Sturm-liouville Problem, turning points, Discrete spec- trum, Asymptotic solutions, Regularized trace formula

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References


I. M. Gelfand,B. M. Levitan, On a Simple Identity for the Characteristic Values of a

Differential Operator of Second Order, Dokl. Akad. Nauk SSSR, Vol.88,(1953),593–596.

I. M. Gelfand, B. M. Levitan, On the determination of adifferential equation from its

spectral function, Amer.Math.Soc.1(1955),253–304.

I. M. Gelfand, On Identities for Eigenvalues of a Differential Operator of Second Order,

Uspekhi Mat. Nauk, Vol.11,1(1956),191–198.

E. Adig¨uzelor, Y. Sezer,The second regularized trace of aself adjoint differential operator

given in a finite interval with buounded operator coefficient, Mathematical and Computer

Modelling,53(2011),553–565.

M. Bayramaglu,H. Sahinturk,Higher order regularized trace formula for the regu-

lar Sturm-Liouville equation continted spectral parameter in the boundary condition,

Appl.Math.Comput.,186(2007),1591–1599

D. Borisov,p. Freitas, Eigenvalue asymptotics inverse problem and trace formula for the

linear damped wave equation, Journal of differntial equations,247(2009),3028–3039.

L. A. Dikii, On a Formula of Gelfand-Levitan, Uspekhi Mat. Nauk, Vol.82(1953), 119–123.

L. A. Dikii,The Zeta Function of an ordinarary Differential Equation on a Finite Inter-

val,IZV.Akad.Nauk SSSR,Vol.19,4(1955),187–200.

B. M. Levitan, Calculation of the Regularized Trace for the Sturm-Liouville Operator, Us-

pekhi Mat. Nauk,Vol.19,1( 1964), 161–165.

M. G. Gasymov, On the Sum of Differences of Eigenvalues of Two Self-Adjoint Operators,

Dokl. Akad. Nauk SSSR, Vol.150,6(1963),1202–1205.

V A. Sadovnichi, V. E.Podolski, Traces of differential operators, ISSN 0012-2661, Differ-

ential Equations,Vol. 45,4 (2009),477–493.

E. Adig¨uzelov, O¨. Baksi,On the regularized trace of the differential equation given in a

finite interval,Sigma Journal of Engineering and Natural Sciences,1(2004),47–55.

R. Z. Chalilova,On regularization of the trace of the Sturm-Liouville operator equa-

tion,Funks,Analiz,Teoriya Funksiyik Pril.Mahaanikala,3(1976),154–161.

F. G. Maksudov, M. Bayramoglu, E. E. Adiguzelov,On a symptotic of spectrum and trace

of high order differential operator with operator coefficient ,Dog˜a-Turkish Journal of Math-

ematics ,Vol.17,2(1993),113–128.

L. V. Bogacher, Trace of the Sturm-Liouville operator with non local boundary condi-

tions,Russ.Mat.Zamelki, Vol.28,3 (1980),379–478.

B.M. Levitan,I. S. Sargsjan, Introduction to spectral theory self adjoint ordinaray differential

operators,Amer Math.Soc.Proc.R.I.(1975),77–81.

S. I. Mitrokhin,Regularized trace formulas for second-order differential operators with dis-

countinuous coefficient,Vestnik Moskov.Univ.Ser.I .Mat.Mekh ,6(1986),3–6

M. Bayramoglu,N. Aslanova,Formula for second regularized trace of aproblem with spectral

parameter dependent boundary condition,Hacettepe Journal of Mathematiccs and Statistics

,Vol.40,5(2011),635–647.

F. Hira,N. Altinisik,The regularized trace of Sturm-Liouville problem with discontinuities at

two points,math.CA,arxiv:1407.5698

Z. F. A. El-Raheem, On some trace formula for a singular Sturm-Liouville operator, Journal

of Pure Math. and Application,Vol.7, 1-2(1996),61–68.

Z. F. A. El-Raheem ,A. H. Nasser The regularized trace formula ofthe spectrum of a Dirich-

let boundary value problem with turning point, Journal of abstract and Applied Analysis

,(2012),1–12.

Z. F. A. El-Raheem,Sh. A. M. Hagag,On the spectral study of singular Sturm-Liouville

problem with sign valued weight,Electronic Journal of Mathematical Analysis and Applica-

tions,Vol.5,2(2017),98–115.

V. A. Marchenko, Sturm-Liouville Operators and Applications,American mathematical so-

ciety,(2011).


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