A unuqueness theorem for Sturm-Lioville operators with eigenparameter dependent boundary conditions

Article Sidebar

PDF
Published: Mar 31, 2012
DOI: https://doi.org/10.5556/j.tkjm.43.2012.1024
Keywords:
Gesztesy-Simon theorem inverse problem eigenparameter dependent boundary condition spectrum.

Main Article Content

Yu-Ping Wang
Department of Applied Mathematics, Nanjing Forestry University, Nan-jing, Jiangsu,China

Abstract

In this paper, we discuss the inverse problem for Sturm- Liouville operators with boundary conditions having fractional linear function of spectral parameter on the finite interval $[0, 1].$ Using Weyl m-function techniques, we establish a uniqueness theorem. i.e., If q(x) is prescribed on $[0,\frac{1}{2}+\alpha]$ for some $\alpha\in [0,1),$ then the potential $q(x)$ on the interval $[0, 1]$ and fractional linear function $\frac{a_2\lambda+b_2}{c_2\lambda+d_2}$  of the boundary condition are uniquely determined by a subset $S\subset \sigma (L)$ and fractional linear function $\frac{a_1\lambda+b_1}{c_1\lambda+d_1}$ of the boundary condition.

Article Details

How to Cite
Wang, Y.-P. (2012). A unuqueness theorem for Sturm-Lioville operators with eigenparameter dependent boundary conditions. Tamkang Journal of Mathematics, 43(1), 145–152. https://doi.org/10.5556/j.tkjm.43.2012.1024
Issue
Vol. 43 No. 1 (2012)
Section
Papers

References

P. A. Binding, P. J. Browne and K. Seddighi, Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Roy. Soc. Edinburgh., 37(1993), 57--72.

C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh, 77a(1977), 293--308.

P. J. Browne and B. D. Sleeman, Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems, 12(1996), 377--381.

N. J. Guliyev, The regularized trace formula for the Sturm-Liouville equation with spectral parameter in the boundary conditions, Proc. Inst. Math. Natl. Alad. Sci. Azerb, 22(2005), 99--102.

Y. P. Wang, An interior inverse problem for Sturm—Liouville operators with eigenparameter dependent boundary conditions, TamKang Journal of Mathmetics, 42(3)(2011), 395--403.

G. Freiling and V. A. Yurko, Inverse problems for Sturm--Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inv. Probl., 26(2010), p. 055003 (17pp.).

R. E. Gaskell, A problem in heat conduction and an expansion theorem, Amer. J. Math., 64(1942), 447--455.

W. F. Bauer, Modified Sturm-Liouville systems, Quart. Appl. Math., 11(1953), 273--282.

H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problem wity mixed given data, SIAM Journal of Applied Mathematics, 34 (1978), 676-680.

R. D. R. Castillo, On boundary conditions of an inverse Sturm-Liouville problem, SIAM. J. APPL. MATH.,50(6)(1990), 1745--1751.

T. Suzuki, Inverse problems for heat equations on compact intervals and on circles, I, J. Math. Soc. Japan., 38(1986)39--65. MR 87f:35241.

L. Sakhnovich, Half inverse problems on the finite interval, Inverse Problems, 17(2001), 527--532.

H. Rostyslav and O. Mykytyuk, Half inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems, 20(5)(2004), 1423--1444.

H. Koyunbakan and E. S. Panakhov, Half inverse problem for diffusion operators on the finite interval, J. Math. Anal. Appl.,326(2007), 1024--1030.

G. S. Wei and H. K. Xu, On the missing eigenvalue problem for an inverse Sturm-Liouville problem, J. Math. Pure Appl., 91(2009), 468--475.

S. A. Buterin, On half inverse problem for differential pencils with the spectral parameter in boundary conditions, Tamkang journal of mathematics, 42(3)(2011), 355--364.

F. Gesztesy and B. Simon, On the determination of a potential from three spectra, Amer Math. Soc. Transl., 189 (1999),85--92.

F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential, II, The case of discrete spectrum, Trans. Amer. Math. Soc., 352 (2000),

--2787.

O. H. Hald, The Sturm-Liouville Problem with symmetric potentials, Acta Math., 141(1978), 262—291.

V. A. Marchenko, Some questions in the theory of one-dimensionnal linear differential operators of the second order ,I, Trudy Moscow Math. Ob$breve{s}breve{c}$. 1(1952),

--420(Russian); Transl. in Amer. Math. Soc. Transl., 101(2) (1973), 1--104. MR 15:315b.

J. R. McLaughlin, Inverse spectral theory using nodal points as data-a uniqueness result, J. Differential Equations, 73(1988), 354--362.

C. F. Yang, Reconstruction of the diffusion operator from nodal data, Z. Natureforsch., 65a.1.(2010) 100-106.

C. T. Shieh and V. A. Yurko, Inverse nodal and inverse spectral problems for discontinuous boundary value problems, J. Math. Anal. Appl., 347(1) (2008), 266--272.

C. K. Law and J. Tsay, On the well-posedness of the inverse nodal problem, Inverse Problems, 17(2001), 1493--1512.

C. L. Shen and C. T. Shieh, An inverse nodal problem for vectorial Sturm-Liouville equation, Inverse Problems, 16(2000), 349--356.

V. A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, VSP, Utrecht: Inverse Ill-posed Problems Ser. 2002.

B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac operators, Dordrecht: Kluwer Academic Publishers, 1990.

G. Freiling and V. A. Yurko, Inverse Sturm-Liouville Problems and their Applications, New York: NOVA Science Publishers, 2001.