Reconstruction of potential function and its derivatives for Sturm–Liouville problem with eigenvalues in boundary condition

Abstract

The purpose of this article is solving inverse nodal problem for Sturm–Liouville equation with a boundary condition depending on spectral parameter. Taking into account Law and Chen's method, we construct the potential function q and its derivatives by using nodal data. We give several lemmas in order to complete proof of the main theorem. Especially, we obtain an explicit formula for potential function and its derivatives from the nodal data by a pointwise limit.

Keywords:

• inverse nodal problem
• reconstruction formula
• Prüfer substitution

AMS Subject Classifications::

• 34B24
• 34A55
• 34B99

1. Introduction

In this article, we study the inverse nodal problem of making use of the position of nodal points (zeros of eigenfunctions) to seek potential function and its derivatives. The problem was first defined by McLaughlin in 1988 1, which is different from the classical inverse spectral theory of Gelfand and Levitan 2. Later, Hald and McLaughlin 3 and Browne and Sleeman 4 showed that only knowledge of the nodal points can determine the potential function of the regular Sturm–Liouville problem. In 5–12, authors concluded about the smoothness of the potential using only nodal data.

We extend the results of Law and Yang 8 to study the Sturm–Liouville problem with a boundary condition depending on the spectral parameter. Especially, we show that for any nonnegative integer N, when q is C^N+1, q^(k)(x) (k = 1, 2, …, N) can be approximated using nodal data.

We turn our attention to the problem (1) (2) (3) where q(x) is a real and integrable function on [0, π], μ and α ≠ 0 are real parameters 13.

Let λ_o(q, α) < λ₁(q, α) < ··· → ∞ be the eigenvalues of (1.1–1.3) and i = 1, 2, …, n − 1, nodal points of the n-th eigenfunction. Let λ_n be the n-th eigenvalue and be i-th nodal point of the n-th eigenfunction y_n. Also let be the i-th nodal domain of the n-th eigenfunction and let be the associated nodal length. Let j_n(x) be the largest index j such that

In Hochtadt's paper 13, a solution for (1.1) shall be sought in the form By this induction, we see that (4) From the above, we can write (5) where tan(λ₀π) = −α and

Δ denotes the difference operator Δa_i = a_i+1 − a_i. Inductively, for k > 1, Δ^ka_i = Δ^k−1a_i+1 − Δ^k−1a_i, and we introduce the difference quotient operator δ:

THEOREM 1.1 11

Assume that q ∈ L¹[0, π], then where μ is a real parameter.

2. Main results

In this section, we give some lemmas and a reconstruction formula for the potential function q and its derivatives of Sturm–Liouville operator with eigenvalues in the boundary condition. The proof of the following lemma is similar to Law's paper 8.

LEMMA 2.1

Suppose that q ∈ L¹ [0, π]. Then, as n → ∞ for the problem (1.1–1.3) (6) (7)

LEMMA 2.2 7

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

Proposition 2.1 8

If q is a continuous function, then

a.	and
b.	for any fixed k, m ∈ ℕ,
c.	where and

LEMMA 2.3 8

If q ∈ C^N[0, π], then for k = 1, …, N, Δ^kl_j = O(n^−(k+3)) as n → ∞ and the order estimate is independent of j.

LEMMA 2.4 8

Let with each where each . Suppose Φ_j = O(n^−v) and q is sufficiently smooth. Then δ^kΦ_j = O(n^−v) for all k ∈ ℕ.

LEMMA 2.5 8

Suppose f ∈ C^N[0, π] and Then δ^kΦ_j = O(n⁻¹) for any k = 0, 1, …, N.

THEOREM 2.1 8

Let Φ_m(x_j) = ψ₁(x_j)ψ₂(x_j)…ψ_m(x_j), where ψ_i(x_j) = x_j+ki and k_i ∈ ℕ ∪ {0}. If q is C^k on [0, π], then

THEOREM 2.2

If q ∈ C^N+1[0, π], then q^(k)(x) = F(λ_n, μ) · δ^kq(x_j) + O(n⁻¹) for k = 0, …, N where and . The order estimate is uniformly valid for compact subsets of [0, π].

Remark

For k = 1, …, N, be a (k + 1) × (k + 1) Vandermonde matrix. It is well known that To prove the Theorem 2.2, we need following lemma.

LEMMA 2.6 8

(8) Next, we consider the following (k + 1) × (k + 1) matrix: After some operations, we obtain Let B be the matrix at the right-hand side. By Lemma 2.4 and Theorem 2.1

Proof of Theorem 2.2

For k = 1, 2, …, N, let

By Rolle's theorem, g(x_j) = g(x_j+1) = ··· = g(x_j+k) = 0 implies that there is some ξ_1,j+i ∈ (x_j+i, x_j+i+1) such that g′ (ξ_1,j+i) = 0. When we repeat the process, we can find that ξ_k,j ∈ (x_j, x_j+k) such that g^(k)(ξ_k,j) = 0. In view of the definition of g, q^(k)(ξ_k,j) det V_k(x_j) = k!. det A. Hence By Lemma 2.6, . Since q ∈ C^k+1, we get

THEOREM 2.3

Suppose that q in (1.1) is C^N+1 on [0, π] (N ≥ 1) and let j = j_n(x) for each x ∈ [0, π]. Then, as n → ∞, and, for all k = 1, 2, …, N,

Proof

The uniform approximation for q is evident. Suppose that q is continuously differentiable on [0, π]. Apply the intermediate value theorem on Proposition (2.1) (c), then there is some such that Hence Applying the mean value theorem, when n is sufficiently large, then (9) Next we obtain q^(k)(x). Then, we consider by Lemma 2.1. If we use Lemma 2.2 and after some straightforward computations, we yield and inserting the above equality in (2.4), we get (10)

If we use a modified Prüfer substitution due to Ashbaugh–Benguria 14 in (1.1), we obtain (11) Integrating (2.6) from x_j to x_j+1 and using Taylor expansion (12) where by Lemma 2.5. Using some trigonometric equalities in (2.7), we obtain Hence using (2.6) again (13) Note that by Lemma 2.5 Then, by Lemma 2.4. . Also, by Taylor expansion where Finally, using integration by parts and we know that Summarizing from (2.8), we obtain Hence where Thus by Lemma 2.4, for any k ≥ 1. Therefore and so for k = 1, 2, …, N (14) If we use the results of Theorems 2.2 and 2.4, we obtain

THEOREM 2.4

Assume that q is C^N+1 on [0, π]. Then, for k = 1, 2, …, N, The estimate is independent of j.

Proof

In view of derivations and the fact that it suffices to show that (15) But Thus, (2.10) follows by Lemma 2.4. If we write (2.10) in (2.9), we get (16)

3. Numerical results

In this section, we compute exact eigenvalues, nodal points and nodal lengths by using (1.5), (2.1) and (2.2), respectively with computer algebra system-Mathematica.

Example 3.1

Let consider the following Sturm–Liouville problem for a = 1 and q(x) = x² ∈ L²[0, π]. (17) (18) (19)

The detailed results for eigenvalues, nodal points and nodal lengths are shown in Tables 1–3. Especially, Table 2 shows that these results are accurate and explicit. We can see that the conditions of oscillation theorem are provided for q(x) = x². In Table 2, it can be seen that nodal points make an oscillation between 0 and π.

Table 1. The first 10 eigenvalues for the problem (3.1–3.3).

n	1	2	3	4	5	6	7	8	9	10
	λ1	λ2	λ3	λ4	λ5	λ6	λ7	λ8	λ9	λ10
	5.3748	4.1874	4.4582	5.0937	5.8749	6.7291	7.6249	8.5468	9.4861	10.4375

Table 2. Numerical values of nodal points according to the eigenvalues for j = 1, 2, …, n − 1 and .

Table 3. Numerical values of nodal lengths according to the eigenvalues for j = 1, 2, …, n − 1 and .

4. Conclusion

In this article, an inverse nodal problem is solved for Sturm–Liouville problem with a boundary condition depending on spectral parameter. Especially, we obtain a reconstruction formula for the potential function and its derivatives. Therefore, we get more general results than classical Sturm–Liouville problem. And finally, we give a numerical example for nodal parameters.