Reconstruction of the Sturm{Liouville operators with a nite number of tranmission and parameter dependent boundary conditions

Mostafa Fallahi, Mohamad Shahriari, Fereshte Shareghi


In this paper, we study discontinuous Sturm--Liouville problems with the eigenvalue parameter linearly contained in boundary conditions and the coefficients are recovered by $(i)$ Weyl function and $(ii)$ spectral data. Further eigenparameter appears in both of the boundary conditions.


Inverse Sturm--Liouville problem, Weyl M--Function, discontinuous and parameter dependent boundary conditions


B. Keskin, A. Sinan and N.Yalcin, Inverse spectral problems for discontinuous Sturm-Liouville operator with eigenvalue dependent boundary conditions, Commun.

Fac. Sci. Univ. Ank. Series A1, 60(1), 2011, 15-25.

N.J. Guliyev, Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions, Inverse probl., 21, 2005, 1315-1330.

J.Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math Zeitschrift, 133, 1973, 301-312.

C.T. Fulton, Two{point boundary value problems with eigenvalue parameter contained

in the boundary conditions , proc. R. Soc. Edinburg, A77, 1977, 293-308.

G. Freiling and V.AYurko, Inverse problem for Sturm{Liouville problems and their

applications, Nova Science, New York, 2001.

B.M. Levitan and I.S. Sargsyan, Introduction to spectral theory, American Mathematical

Society, Providence, RI, USA, 1975.

D.G. Shepelsky, The inverse problem of reconstruction of the medium conductivity in a

class of discontinuous and increasing functions, Spectral operator theory and related

topics, 209{232, Advances in Soviet Math., 19, Amer. Math. Soc., providence, RI.

E.M. Russakovsk, Operator treatment of boundary problems with spectral parameters

entering via polynomials in the boundary conditions, Funct. Anal. Appl., 9, 1975,


E.A. Codington and N. Levinson, Theory of ordinary dierential equations, McGraw-

Hill, New York, NY, USA, 1955.

R. Kh. Amirov, On Sturm{Liouville operators with discontinuity conditions inside

an interval, J. Math. Anal. Appl., 317, 2006, 163{176.

I.M. Gelfand and B.M. Levitan, On the determination of a dierential equation from

its spectral function Amer, Math. Soc. Transl. Ser., 21, 1955, 253{304.

M. Shahriari, Inverse Sturm{Liouville problem with eigenparameter dependent boundary

and transmission conditions, Azerb. J. Math., 4(2), 2014, 16{30.

M. Shahriari, Inverse Sturm{Liouville problems with a spectral parameter in the

boundary and transmission condition, Sahand Commun. in Math. Anal., 3(2), 2016,


M. Shahriari, A. Jodayree Akbarfam, and G.Teschl, Uniqueness for Inverse Sturm{

Liouville problems with a nite number of transmission conditions, J. Math. Anal.

Appl., 395, 2012, 19{29.

N.B. kerimov and V.S. Mirzoev, On the basis properties of one spectral parameter in

the boundary conditions, Sibirsk. Mat. Zh., 33, 1997, 116{120,

P.J. Browne and B.D. Sleeman, A uniqueness theorem for inverse eigenparameter dependent

Sturm{Liouville problems, Inverse Probl., 13(6), 1997, 1453{1462.

V.A.Yurko, Reconstruction of singular non-self-adjoint dierential operators with a

singularity inside an interval, Di. Equat., 38(5), 2002, 678{694.

V.A.Yurko, Inverse spectral problems for linear dierential operators and their applications,

Gordon and Breach, New York, 2000.

O.H. Hald, Discontinuous inverse eigenvalue problems, Comm. Pure Appl. Math., 37,

, 539{577.

V.A. Marchenko, The Sturm{Liouville operators and their applications(Russian),

(Kiev: Naukova Dumka) 1977; English transl. (Basel: Birkhauser Verlag, 1986).

R.S. Anderssen, The eect of discontinuities in density and shear velocity on the

asymptotic overtone structure of to rational eigenfrequencies of the earth, Geophys.

J. R. astr. Soc., 50, 1997, 303{309.

G.Teschl, Mathematical methods in Quantum mechanics; with applications to

Schrodinger operators, Graduate studies in mathematics, Amer. Math. Soc., Rhode

Island, 2009.

Y. Guo and G.Wei, On the reconstruction of the Sturm{Liouville problems with spectral

parameter in the discontinuity conditions, Results Math. 65(3), 2014, 385{398.


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