- Open Access
Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation
© Koyunbakan; licensee Springer. 2013
Received: 15 October 2013
Accepted: 19 November 2013
Published: 12 December 2013
The Erratum to this article has been published in Boundary Value Problems 2014 2014:222
In this study, the inverse nodal problem is solved for p-Laplacian Schrödinger equation with energy-dependent potential function with the Dirichlet conditions. Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by using Prüfer substitution. Especially, an explicit formula for a potential function is given by using nodal lengths. Results are more general than the classical p-Laplacian Sturm-Liouville problem. For the proofs, methods previously developed by Law et al. and Wang et al., in 2009 and 2011, respectively, are used. In there, they solved an inverse nodal problem for the classical p-Laplacian Sturm-Liouville equation with eigenparameter boundary conditions.
The determination of the form of a differential operator from spectral data associated with it has enjoyed close attention from a number of authors in recent years. One such operator is the Sturm-Liouville operator. In the typical formulation of the inverse Sturm-Liouville problem, one seeks to recover both q and constants by giving the eigenvalues with another piece of spectral data. These data can take several forms, leading to many versions of the problem. Especially, the recent interest is a study by Hald and McLaughlin [9, 10] wherein the given spectral information consists of a set of nodal points of eigenfunctions for the Sturm-Liouville problems. These results were extended to the case of problems with eigenparameter-dependent boundary conditions by Browne and Sleeman . On the other hand, Law et al., Law and Yang  solved the inverse nodal problem of determining the smoothness of the potential function q of the Sturm-Liouville problem by using nodal data. In the past few years, the inverse nodal problem of Sturm-Liouville problem has been investigated by several authors [11, 14–16].
for arbitrary . These functions are known as generalized sine and cosine functions and for become sine and cosine.
Now, we present some further properties of for deriving more detailed asymptotic formulas. These formulas are crucial in the solution of our problem.
According to the Sturm-Liouville theory, the zero set of the eigenfunction corresponding to is called the nodal set and is defined as the nodal length of . Using the nodal data, some uniqueness results, reconstruction and stability of potential functions have been obtained by many authors [9, 11, 14–16, 18].
where and are real-valued functions, is a constant, and λ is the spectral parameter.
In this paper, the function r is known a priori and we try to construct the unknown function q by the dense nodal points in the interval considered.
This equation is known as the diffusion equation or quadratic of differential pencil. Eigenvalue equation (1.6) is important for both classical and quantum mechanics. For example, such problems arise in solving the Klein-Gordon equations, which describe the motion of massless particles such as photons. Sturm-Liouville energy-dependent equations are also used for modelling vibrations of mechanical systems in viscous media (see ). We note that in this type of problem the spectral parameter λ is related to the energy of the system, and this motivates the terminology ‘energy-dependent’ used for the spectral problem of the form (1.6). Inverse problems of quadratic pencil have been solved by many authors in the references [15, 16, 18, 20–27].
and associated eigenfunctions are denoted by .
This paper is organized as follows. In Section 2, we give asymptotic formulas for eigenvalues, nodal points and nodal lengths. In Section 3, we give a reconstruction formula for differential pencil by using nodal parameters.
2 Asymptotic estimates of nodal parameters
In this section, we study the properties of eigenvalues of p-Laplacian operator (1.3) with Dirichlet conditions (1.4). For this, we introduce Prüfer substitution. One may easily obtain similar results for Neumann problems.
Inserting these values in (2.5) and after some straightforward computations, we obtain (2.4). □
3 Reconstruction of a potential function in the differential pencil
This completes the proof. □
Conclusion 3.2 In Theorem 2.1, Theorem 2.2, Theorem 2.3 and Theorem 3.1, taking , we obtain results of the Sturm-Liouville problem given in .
Conclusion 3.3 In Theorem 2.1, Theorem 2.2, Theorem 2.3 and Theorem 3.1, taking , we obtain the results of an inverse nodal problem for differential pencil .
The author would like to thank the referees for valuable comments and suggestions on improving this paper.
- Law CK, Lian WC, Wang WC: Inverse nodal problem and Ambarzumyan theorem for the p -Laplacian. Proc. R. Soc. Edinb. A 2009, 139: 1261-1273. 10.1017/S0308210508000851MathSciNetView ArticleGoogle Scholar
- Binding P, Drábek P: Sturm-Liouville theory for the p -Laplacian. Studia Sci. Math. Hung. 2003, 40: 373-396. 10.1556/SScMath.40.2003.4.1Google Scholar
- Binding PA, Rynne BP: Variational and non-variational eigenvalues of the p -Laplacian. J. Differ. Equ. 2008, 244: 24-39. 10.1016/j.jde.2007.10.010MathSciNetView ArticleGoogle Scholar
- Brown BM, Eastham MSP: Eigenvalues of the radial p -Laplacian with a potential on . J. Comput. Appl. Math. 2006, 208: 111-119.MathSciNetView ArticleGoogle Scholar
- Del Pino M, Drábek P, Manasevich R: The Fredholm alternatives at the first eigenvalue for the one-dimensional p -Laplacian. J. Differ. Equ. 1999, 151: 386-419. 10.1006/jdeq.1998.3506View ArticleGoogle Scholar
- Drábek P: On the generalization of the Courant nodal domain theorem. J. Differ. Equ. 2002, 181: 58-71. 10.1006/jdeq.2001.4070View ArticleGoogle Scholar
- Walter W: Sturm-Liouville theory for the radial p -operator. Math. Z. 1998, 227: 175-185. 10.1007/PL00004362MathSciNetView ArticleGoogle Scholar
- Wang WC, Cheng YH, Lian WC: Inverse nodal problem for the p -Laplacian with eigenparameter dependent boundary conditions. Math. Comput. Model. 2011, 54: 2718-2724. 10.1016/j.mcm.2011.06.059MathSciNetView ArticleGoogle Scholar
- Hald OL, McLaughlin JR: Solutions of inverse nodal problems. Inverse Probl. 1989, 5: 307-347. 10.1088/0266-5611/5/3/008MathSciNetView ArticleGoogle Scholar
- McLaughlin JR: Inverse spectral theory using nodal points as data - a uniqueness result. J. Differ. Equ. 1988, 73: 354-362. 10.1016/0022-0396(88)90111-8View ArticleGoogle Scholar
- Browne PJ, Sleeman BD: Inverse nodal problem for Sturm-Liouville equation with eigenparameter depend boundary conditions. Inverse Probl. 1996, 12: 377-381. 10.1088/0266-5611/12/4/002MathSciNetView ArticleGoogle Scholar
- Law CK, Shen CL, Yang CF: The inverse nodal problem on the smoothness of the potential function. Inverse Probl. 1999, 15: 253-263. 10.1088/0266-5611/15/1/024MathSciNetView ArticleGoogle Scholar
- Law CK, Yang CF: Reconstructing the potential function and its derivatives using nodal data. Inverse Probl. 1998, 14: 299-312. Addendum 14, 779-780 (1998) 10.1088/0266-5611/14/2/006MathSciNetView ArticleGoogle Scholar
- Buterin SA, Shieh CT: Incomplete inverse spectral and nodal problems for differential pencil. Results Math. 2012, 62: 167-179. 10.1007/s00025-011-0137-6MathSciNetView ArticleGoogle Scholar
- Koyunbakan H, Yılmaz E: Reconstruction of the potential function and its derivatives for the diffusion operator. Z. Naturforsch. A 2008, 63: 127-130.Google Scholar
- Yang XF: A solution of the inverse nodal problem. Inverse Probl. 1997, 13: 203-213. 10.1088/0266-5611/13/1/016View ArticleGoogle Scholar
- Lindqvist P: Some remarkable sine and cosine functions. Ric. Mat. 1995, XLIV(2):269-290.MathSciNetGoogle Scholar
- Koyunbakan H: A new inverse problem for the diffusion operator. Appl. Math. Lett. 2006, 19(10):995-999. 10.1016/j.aml.2005.09.014MathSciNetView ArticleGoogle Scholar
- Jaulent M, Jean C: The inverse s -wave scattering problem for a class of potentials depending on energy. Commun. Math. Phys. 1972, 28: 177-220. 10.1007/BF01645775MathSciNetView ArticleGoogle Scholar
- Gasymov MG, Guseinov GS: The determination of a diffusion operator from the spectral data. Dokl. Akad. Nauk Azerb. SSR 1981, 37(2):19-23.MathSciNetGoogle Scholar
- Guseinov GS: On spectral analysis of a quadratic pencil of Sturm-Liouville operators. Sov. Math. Dokl. 1985, 32: 859-862.Google Scholar
- Hryniv R, Pronska N: Inverse spectral problems for energy dependent Sturm-Liouville equations. Inverse Probl. 2012., 28: Article ID 085008Google Scholar
- Nabiev IM: The inverse quasiperiodic problem for a diffusion operator. Dokl. Math. 2007, 76: 527-529. 10.1134/S1064562407040126MathSciNetView ArticleGoogle Scholar
- Wang YP, Yang CF, Huang ZY: Half inverse problem for a quadratic pencil of Schrödinger operators. Acta Math. Sci. 2011, 31(6):1708-1717.MathSciNetGoogle Scholar
- Yang CF: Trace formulae for the matrix Schrödinger equation with energy-dependent potential. J. Math. Anal. Appl. 2012, 393: 526-533. 10.1016/j.jmaa.2012.03.003MathSciNetView ArticleGoogle Scholar
- Koyunbakan H: Inverse problem for quadratic pencil of Sturm-Liouville operator. J. Math. Anal. Appl. 2011, 378: 549-554. 10.1016/j.jmaa.2011.01.069MathSciNetView ArticleGoogle Scholar
- Yurko VA Inverse and Ill-Posed Problems Series. In Method of Spectral Mappings in the Inverse Problem Theory. VSP, Utrecht; 2002.View ArticleGoogle Scholar
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