Reconstruction formula for the potential function of Sturm–Liouville problem with eigenparameter boundary condition

Abstract

It is known that the uniqueness of potential function of the Sturm–Liouville problem can be shown from the nodal points. In this article, we solve the inverse nodal problem of the reconstruction of the potential function q from the nodal data by a pointwise limit. We show that this convergence is in the L¹. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method for the Sturm–Liouville problem depending on eigenparameter boundary conditions.

Keywords:

• Sturm–Liouville problem
• nodal points
• reconstruction formula

AMS Subject Classifications::

• 34B24
• 34A55

1. Introduction

Inverse problems of spectral theory have been an important part of the research in mathematical physics. The development of the inverse problem, its close relative, is quite different. In general, a set of eigenvalues is not sufficient to determine the operator. It needs more data that are spectral function, norming constants, etc., to determine the operator. For example, two sets of eigenvalues 1, one set of eigenvalues plus a symmetric potential function 2 and a spectral function 3. On the other hand, some questions relating to the Sturm–Liouville problem were solved by Gesztesy and Simon 4,5 and Malamud 6.

Recently, a new class of inverse problems has attracted the attention of researchers. This is the so-called inverse nodal problem. The inverse nodal problem was initiated by McLaughlin 7, which is different from the classical inverse spectral theory of Gelfand and Levitan 3. She found that the nodal set of the Dirichlet problem for the Sturm–Liouville equation can determine q up to a constant. Later, Hald and McLaughlin 8 and Browne and Sleeman 9 showed that only the knowledge of nodal points can determine the potential function of the regular Sturm–Liouville problem. Yang 10 showed that this uniqueness result is valid for any q. In the past few years, the inverse nodal problem of the regular and singular Sturm–Liouville problem has been investigated by several authors 7–9,11,12.

Consider the following equation: with the boundary conditions where q(x) is real and integrable on [0, π], and μ and a ≠ 0 are real parameters. It is well known that the above problem has only real and simple eigenvalues 13.

In Hochstadt's paper 13, a solution of (1.1) shall be sought in the form By this induction, we see that so that By using (1.3) we get 13 where tan λ_oπ = −a and We observe that μ dependence enters into the terms, vanishing like .

Let λ_o(q, a) < λ₁(q, a) < ··· →∞ be the eigenvalues of the (1.1)–(1.3) and i = 1, 2, …, n − 1, be nodal points of the nth eigenfunction. Let λ_n be the nth eigenvalue and be the ith nodal point of the nth eigenfunction y_n. Also, let be the ith nodal domain of the nth eigenfunction and be the associated nodal length. 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 to this, it will be bound 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.

We define the function j_n(x) on (0, π) by Hence fix x and n, j = j_n(x) implies

The inverse nodal problem was first studied by McLaughlin 7 and Hald and McLaughlin 8, then this method was extended in 14,15. In this article, these results are expanded. In other words, for the Sturm–Liouville problem which includes μ parameter and a general boundary condition, a reconstruction formula for q is given. Our main theorems are as follows.

THEOREM 1.1

Suppose that q ∈ C¹ on [0, π], then where μ is as expressed in (1.1).

We note that with the asymptotic expression for λ_n in Theorem 1.1 implies that , where F_n is determined only by the nodal data and the constant in the form and μ is as expressed in (1.1).

THEOREM 1.2

F_n converges to q in L¹.

2. Main results

In this section, some lemmas which are necessary for proof of the Theorem 1.1 and Theorem 1.2 are given.

LEMMA 2.1

Assume that q ∈ L¹ (0, π). Then as n → ∞

Proof

From (1.4), we know that If y(x, λ) = 0, then as long as cos λx is not close to zero, then Now, we take λ = λ_n and . Since the Taylor's expansion for the arctangent function is given by then Therefore The nodal length is

LEMMA 2.2

Suppose that f ∈ L¹(0, π). Then for almost every x ∈ (0, π), with j = j_n(x)

Proof

For f ∈ L¹(0, π), almost everywhere. Then, given ξ > 0, when n is sufficiently large and for almost every x ∈ (0, π) This proves the lemma.▪

Proof of Theorem 1.1

When we consider (2.2) in the form so that By Lemma 2.2 then or It remains to be shown that for almost every x ∈ (0, π), tends to zero as n → ∞.

We get a sequence of continuous functions q_k which converges to q in L¹(0, π). Then q_k has a subsequence converging to q almost everywhere in (0, π). For simplicity, we call this subsequence q_k. We take any x such that q_k(x) converges to q(x). Then for a given ϵ > 0, we can fix a large k such that |q_k(x) − q(x)| < ϵ. Hence, Now, by Lemma 2.1 and this term tends to zero for n → ∞. By (1.6) the first term A_n satisfies, when n is sufficiently large, On the other hand, Because q_k is continuous, this term is arbitrarily small for every x ∈ (0, π). For n → ∞, Then, D_n → 0. Hence we conclude that This proves the theorem.

LEMMA 2.3

We take that the sequence f_k ∈ [0, π] converges to f ∈ L¹, then, for any large enough n, with j = j_n(x), as k → ∞

Proof

By Lemma 2.1 and the observation that the integral is constant on any nodal interval and for k → ∞ this term converges to zero.▪

LEMMA 2.4

Suppose that q ∈ L¹ (0, π), then, as n → ∞ with j = j_n(x)

Proof

First, we show that if q is continuous on [0, π] the result satisfies. Let By using the intermediate value theorem, there exists ζ ∈ (a, x) such that If x is close enough to a, the difference can be arbitrarily small. Then, for all ϵ > 0, when n is large enough, with j = j_n(x) In the above process, we assume that The estimate also holds if Hence if q ∈ C[0, π], then converges to q(x) uniformly on [0, π]. Thus can be arbitrarily small. Since C[0, π] is dense in L¹(0, π), for any q ∈ L¹(0, π) there exists a sequence q_k ∈ C [0, π] convergent to q in L¹(0, π). Hence fix n sufficiently large,

By the above process and Lemma 2.3, when k is large enough, the first two terms are arbitrarily small. Hence, as k → ∞

Proof of Theorem 1.2

When we consider the value of F_n, we obtained that It sufficies to show that as n → ∞ By using Lemma 2.1 we have Hence, we only need to prove that for n → ∞ and From Lemma 2.3, the first limit holds and the second limit also holds. On the other hand, the sequence of functions converges to 0 for almost every x ∈ (0, π). Furthermore, and Then, we may apply the Lebesque dominated convergence theorem to show that (2.3) is valid. The proof is completed.

Acknowledgements

This work was supported in part by the Firat University, Scientific Research Project Unit under grant number: 1540.