A uniqueness theorem for inverse nodal problem

Abstract

In this article, it is found that the asymptotic formulas for nodal points and nodal length for the differential operators having singularity type at the points 0 and π, it is shown that the potential function can be determined from the positions of the nodes for the eigenfunctions.

Keywords:

• Sturm--Liouville problem
• Eigenfunction
• Nodal point

Keywords:

• 2000 Mathematics Subject Classifications: 34B24
• 34B99

1. Introduction

Inverse problems of spectral analysis imply the reconstruction of a linear operator from some or other of its spectral characteristics. Such characteristics are spectra (for different boundary conditions), normalizing constants, spectral functions, scattering data etc. An early important result in this direction, which gave vital impetus for the further development of inverse problem theory, was obtained 1. At present, inverse problems are studied for certain special classes of ordinary differential operators. Inverse problems from spectra are the most simple in their formulation and well-studied in 18,24. An effective method of constructing a regular and singular Sturm--Liouville operator from a spectral function or from two spectra are given 9,13,19,22,23.

We note that the detail of inverse problem for singular equations are given in the monographs and references therein 3,5,6,8,12,17.

In some recent interesting works 2,11, Browne and Sleeman, and Hald and Mclaughlin have taken a new approach to inverse spectral theory for the Sturm--Liouville problem. The novelty of this work lies in the use of nodal points as the given spectral data. In recent years, inverse nodal problems have been studied by several authors 2,4,7,21,25 etc.

In this article, we study inverse spectral theory for a singular Sturm--Liouville problem using a new kind of spectral data.

Consider the Sturm--Liouville problem: (1) (2) (3) where q is integrable and h and H are finite numbers.

Let λ_n be nth eigenvalue and be jth nodal point of the nth eigenfunction y_n. It is well known that the problem (1.1–1.3) has only real and simple eigenvalues . The corresponding eigenfunction y_n has n − 1 nodal points in (0,1). So, , j=1,2,…,n-1. Let and be the nodal length.

Before giving the main results of this article, we mention some properties of the classical Legendre equation. This equation is written as we let then y satisfies where λ =n(n+1). For and n sufficiently large, we conclude that Legendre functions are 14 (4) We consider the problem (5) (6) (7) where h is finite number and  ϵ positive number. We accept that the solution of the problem (1.5–1.7) is 16 (8) In this case, the kernel K(x, t) satisfies the equation and the boundary conditions Let be the eigenvalues of this problem (1.5–1.7) and , j=1,2,…,n-1 be nodal points of the nth eigenfunction. It is shown that the set of all nodal points is dense in (0,π) ; in fact, a judicious choice of one nodal point for each y_n, n>1 also forms a dense set in (0,π). The simplest method for choosing a dense subset of the nodes is to choose all the eigenfunctions. In addition, it will be found that the nodes for y_n are roughly equally spaced. Hence, we can choose the first node in and the second in The third, fourth, fifth and sixth nodes should lie in and and so on. This method gives a dense set of nodes and it is not even necessary to choose the next node from the next eigenfunction and any finite amount of nodes can be deleted.

2. Main results

In this section, our purpose is to develop asymptotic expressions for the points and at which y_n, the eigenfunction corresponding to the eigenvalue λ_n of the problem (1.4–1.6) vanishes.

THEOREM 2.1 

We consider the equation (9) with the boundary conditions (10) (11) Then, the nodal points and nodal length of the problem (2.1–2.3) satisfy (12) (13)

Proof 

Zeros of (2.3) for  λ are eigenvalues of the problem (2.1–2.3). So, the asymptotic expression of eigenvalues is 15 (14) for constant a, where m

In fact, for positive  λ the estimate (1.4) assumes the form Furthermore, differentiating the relation (1.4) with respect to x and inserting the values of the ϕ(x,λ) and into the boundary condition (1.7) after some straightforward computations, we get The above asymptotic formula can be improved substantially. Indeed, differentiating (1.8) with respect to x and inserting in (1.7), we obtain after simple transformations then we yield (2.6). We recall that the above method was used by Levitan and Sargsjan 20 for regular Sturm--Liouville problem.

The eigenfunctions of the problem (2.1–2.3) are in the asymptotic form 14 Hence, we use the classical estimate where M is a constant. Thus ϕ(x,λ) will vanish in the intervals whose end points are solution to This equation can also be written as Using the Taylor expansion for arc cos(M₁), then we get or Inserting (2.6) in the last equation, we get The nodal length is The proof is complete.

Now, we will prove a uniqueness theorem. It says that the potential function q(x) for a singular Sturm--Liouville problem is uniquely determined by a dense subset of the nodes. We mentioned that this theorem was given for regular Sturm--Liouville problems by McLaughlin 21, Hald and McLaughlin 11 and Browne and Sleeman 2.

THEOREM 2.2 

Suppose that q is integrable. Then, h and are uniquely determined by any dense set of nodal points.

Proof 

Assume that we have two problems of the type (2.1–2.3) with the h, and q, . Let the nodal points , satisfying form a dense set in We take solutions of (2.1–2.3) w_n for and for (. It follows from (2.1) that (15) Recall that are uniformly bounded in n and the are uniformly bounded in n and . We select a subsequence of nodes from the dense set. To show that we integrate both sides of (2.7) from to  π and choose a subsequence that tends to π, and we see that From this results we can say

Let and integrate both sides of (2.7) from 0 to From the asymptotic forms of and λ_n, we have We take a sequence accumulating at an arbitrary . Hence, this holds for all x. We can therefore conclude that is uniquely determined by a dense set of nodes. This completes the proof.

COROLLARY 

For the problem (2.1–2.3), the potential q is uniquely determined by a dense set of nodes and the constant .

Proof 

Suppose that . Since it follows that . Hence, we can conclude from Theorem 2.2 that almost everywhere on

3. Conclusion

We mention that the above problem is solved by several authors for regular differential operator. In this work, a uniqueness theorem of potential function for singular differential operator is given. Morever, when h is finite number and q is integrable function, nodal points and nodal length can be estimated for this problem.