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.
MSC: 34A55, 34L20.
Keywords:Prüfer substitution; inverse nodal problem; p-Laplacian equation
Consider the following p-Laplacian eigenvalue problem for
with the boundary conditions
where is a real-valued function, is a constant, and λ is the spectral parameter . Equation (1.1) is also known as a one-dimensional p-Laplacian eigenvalue equation. Note that when , equation (1.1) becomes a Sturm-Liouville equation as
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].
When , consider the problem
The eigenvalues of this problem were given as 
and an associated eigenfunction is denoted by . and are periodic functions satisfying the identity
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.
(a) For ,
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].
Consider the p-Laplacian eigenvalue problem
with the Dirichlet conditions
or with the Neumann boundary conditions
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.
For , equation (1.5) becomes
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].
As in the p-Laplacian Sturm-Liouville problem, for , eigenvalues of the problem given by (1.3), (1.4) are of the form
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.
We define a modified Prüfer substitution
Differentiating both sides of equation (2.2) with respect to x and applying Lemma 1.1, one obtains that
Theorem 2.1The eigenvalues of the Dirichlet problem given in (1.3), (1.4) have the form
Proof For problem (1.3), (1.4), let , and . Firstly, we integrate both sides of (2.3) over the interval :
Using the identity
and Lemma 1.1(b), we get
Then, using integration by parts, we have
and when ,
Similarly, one can show that
Inserting these values in (2.5) and after some straightforward computations, we obtain (2.4). □
Theorem 2.2For problem (1.3), (1.4), the nodal points expansion satisfies
Proof Let and integrating (2.3) from 0 to , we have
By using the estimates of eigenvalues as
Theorem 2.3As ,
Proof For large , integrating (2.3) on and then
By Lemma 1.1 and a similar process to that used in Theorem 2.1, we obtain that
where and . Similarly, one can show that
Inserting this value in (2.7), we obtain
and by Theorem 2.1,
3 Reconstruction of a potential function in the differential pencil
Theorem 3.1Let , and assumerthat on the interval is given a priori. Then
Proof Applying the mean value theorem for integrals to (2.6), with fixed n, there exists , we obtain
Considering (2.6), we can write that for ,
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 declares that they have no competing interests.
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. 139, 1261–1273 (2009). Publisher Full Text
Binding, P, Drábek, P: Sturm-Liouville theory for the p-Laplacian. Studia Sci. Math. Hung.. 40, 373–396 (2003). Publisher Full Text
Binding, PA, Rynne, BP: Variational and non-variational eigenvalues of the p-Laplacian. J. Differ. Equ.. 244, 24–39 (2008). Publisher Full Text
Del Pino, M, Drábek, P, Manasevich, R: The Fredholm alternatives at the first eigenvalue for the one-dimensional p-Laplacian. J. Differ. Equ.. 151, 386–419 (1999). Publisher Full Text
Drábek, P: On the generalization of the Courant nodal domain theorem. J. Differ. Equ.. 181, 58–71 (2002). Publisher Full Text
Walter, W: Sturm-Liouville theory for the radial p-operator. Math. Z.. 227, 175–185 (1998). Publisher Full Text
Wang, WC, Cheng, YH, Lian, WC: Inverse nodal problem for the p-Laplacian with eigenparameter dependent boundary conditions. Math. Comput. Model.. 54, 2718–2724 (2011). Publisher Full Text
Hald, OL, McLaughlin, JR: Solutions of inverse nodal problems. Inverse Probl.. 5, 307–347 (1989). Publisher Full Text
McLaughlin, JR: Inverse spectral theory using nodal points as data - a uniqueness result. J. Differ. Equ.. 73, 354–362 (1988). Publisher Full Text
Browne, PJ, Sleeman, BD: Inverse nodal problem for Sturm-Liouville equation with eigenparameter depend boundary conditions. Inverse Probl.. 12, 377–381 (1996). Publisher Full Text
Law, CK, Shen, CL, Yang, CF: The inverse nodal problem on the smoothness of the potential function. Inverse Probl.. 15, 253–263 (1999). Publisher Full Text
Addendum 14, 779-780 (1998)Publisher Full Text
Buterin, SA, Shieh, CT: Incomplete inverse spectral and nodal problems for differential pencil. Results Math.. 62, 167–179 (2012). Publisher Full Text
Yang, XF: A solution of the inverse nodal problem. Inverse Probl.. 13, 203–213 (1997). Publisher Full Text
Koyunbakan, H: A new inverse problem for the diffusion operator. Appl. Math. Lett.. 19(10), 995–999 (2006). Publisher Full Text
Jaulent, M, Jean, C: The inverse s-wave scattering problem for a class of potentials depending on energy. Commun. Math. Phys.. 28, 177–220 (1972). Publisher Full Text
Nabiev, IM: The inverse quasiperiodic problem for a diffusion operator. Dokl. Math.. 76, 527–529 (2007). Publisher Full Text
Yang, CF: Trace formulae for the matrix Schrödinger equation with energy-dependent potential. J. Math. Anal. Appl.. 393, 526–533 (2012). Publisher Full Text
Koyunbakan, H: Inverse problem for quadratic pencil of Sturm-Liouville operator. J. Math. Anal. Appl.. 378, 549–554 (2011). Publisher Full Text