The Eigenfunction Expansion of singular Sturm-Liouville problem with sign-valued weight

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


In the present paper,~we investigate an eigenfunction expansion associated to a singular Sturm-Liouville problem in finite interval with the sign valued weight by two ways,~first by standard method,~second by proving its convergence in some metric space $L_{2}(0,\pi;\rho(x))$.

Full Text:



W. N. Everitt,H. Kalf.Weyl’s work on singular Sturm-Liouville operators,

London Mathematical Society Lecture Note Series 354,(2008).

H. Weyl,Uber gewohnliche Differentialgleichungen mit singularen Stellen und

die zugehorigen Entwicklungen willkurlicher Functionen, Math. Ann. 68

(1910), 220 – 269.

M. H. Stone, Linear Transformations in Hilbert Space, American Mathematical

Society,New York,(1932).

K. Kodaira,The eigenvalue problem for ordinary differential equations of the

second order and Heisenbergs theory of S-matrices, Amer. J. Math.71 (1949),


E. C. Titchmarsh,Eigenfunction Expansions associated with Second Order

Differential Equations, Vol.I, 2nd edn., Oxford,( 1962).

E. C. Titchmarsh,Eigenfunction Expansions associated with Second Order

Differential Equations, Vol. II, Oxford, (1958).

W. O. Amrein,A.M.Hinz ,D.P.Pearson,Sturm-Liouville Theory:Past and

Present ,Springer ,New York,NY,USA,(2005).

Sh. Alemov,on the Tikhonov work about the inverse problem for the SturmLiouville


Z. F. A. El-Raheem A. H. Nasser,The eigenfunction expansion for a Dirichlet

problem with explosive factor,J.Abstract Appl Anal,Vol. 16 (2011), Article ID

B. M. Levitan,Inverse Sturm-Liouville Problems,VNU Press,(1987).

Z. F. A. El-Raheem ,A. H. Nasser,On the spectral property of a Dirichlet

problem with explosive factor, J Appl Math Comput.,Vol. 138,2-3(2003), 355-


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

problem with sign valued weight.,Electronic Journal of Mathematical

Analysis and Applications,Vol.52 (2017),98–115.

M. G. Gasmov,A. Sh. KakhramanovS.K.Petrosyan,On the spectral theory of

linear differential operators with discontinuous coefficient ,Akademiya Nauk

Azerbaˇidzhanskoˇi SSR.Doklady,Vol.43,3(1987),13–16.

V. A. Marchenko,Sturm-Liouville Operators and Applications, AMS, (2011).


  • There are currently no refbacks.