Reconstruction for the spherically symmetric speed of sound from nodal data

Abstract

In this article, we consider the inverse nodal problem for the Sturm–Liouville operator arisen from acoustic scattering problems. We show that a dense subset of nodal points can uniquely determine the potential on the finite interval [0, 1] and coefficient a of the boundary condition and provide a constructive procedure for the solution of the inverse nodal problem. A corollary of this theorem is to reconstruct the spherically symmetric speed of sound on the interval [0, b] for acoustic scattering problems from nodal data.

Keywords:

• inverse nodal problem
• Sturm–Liouville problem
• potential
• spherically symmetric speed of sound

AMS Subject Classifications::

• 34A55
• 34B24
• 47E05

1. Introduction

Inverse nodal problem is to reconstruct operators from the given nodes (zeros) of their eigenfunctions. In 1988, McLaughlin 1 discussed the inverse nodal problem for Sturm–Liouville operators and showed that a dense subset of nodal points of eigenfunctions is sufficient to determine the potential of the Sturm–Liouville operator. Then, Hald and McLaughlin 2 gave some numerical schemes for the reconstruction of the potential. Later, one of such inverse nodal problems for Sturm–Liouville operators was considered by numerous authors 3–12. And inverse nodal problems for diffusion operators, such as vectorial Sturm–Liouville equations; discontinuous boundary value problems; Dirac operators; Sturm–Liouville equations on graphs, were addressed by many authors 13–19. Inverse nodal problem has many applications in mathematics and natural sciences 19,20. Browne and Sleeman 3 considered the inverse nodal problem for Sturm–Liouville equations with eigenparameter dependent on boundary condition. Later, Panakhov et al. 11 discussed the inverse nodal problem for the Sturm–Liouville operator with eigenparameter in one boundary condition. In 2010, Chernozhukova and Freiling 21 established a uniqueness theorem for inverse spectral problems depending nonlinearly on the spectral parameter. Then, using the method of spectral mappings, Freiling and Yurko 22 addressed the inverse problems for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter and obtained three uniqueness theorems on the potential q(x) from the Weyl function, or from discrete spectral data or from two spectra and presented constructive procedures for these problems. Later, Yang (C.F.) and Yang (X.P.) 12 explored the inverse nodal problem for Sturm–Liouville operators with boundary conditions polynomially dependent on the spectral parameter from nodal points and reconstructed the potential q(x) and coefficient functions of the boundary conditions. Motivated by the above results, in this article, we consider the inverse nodal problem for Sturm–Liouville operators with one boundary condition having transcendental functions on the spectral parameter.

Consider the following Sturm–Liouville operator L: L(q, a) defined by(1) with boundary conditions(2) and(3) where 0 ≤ a < 1 or a > 1, q(x) is a real-valued function and q ∈ L²[0, 1].

Denote the function ω(λ) byClearly, ω(λ) is an entire function of order in λ. λ_n(n = 1, 2, …) is a zero of ω(λ) = 0, where λ_n is called as n-th eigenvalue for the Sturm–Liouville operator L(q, a). Since the boundary condition (1.3) is different from the classical boundary condition, inverse spectral problems for the Sturm–Liouville operator L(q, a) can be little discussed. To the best of our knowledge, inverse nodal problem and the trace formula for the Sturm–Liouville operator L(q, a) have not been considered.

In this article, by using the nodal points as data, we discuss the inverse nodal problem for the Sturm–Liouville operator L(q, a). We show that a dense subset of nodal points can uniquely determine the potential q(x) on the interval [0, 1] and coefficient a of the boundary condition and provide a constructive procedure for the solution of the inverse nodal problem. We still establish a uniqueness theorem for the spherically symmetric speed of sound n(r) on the interval [0, b] and present a reconstruction procedure for the spherically symmetric speed of sound n(r). And the oscillation theory for the Sturm–Liouville operator L(q, a) is also obtained, which is a generalization of the classical Sturm–Liouville problem (see Lemma 2.2). We will find the trace formula for the Sturm–Liouville operator L(q, a) in another paper.

Next, we describe the actual background of the Sturm–Liouville operator L(q, a). Acoustic scattering problem was addressed by a number of authors 23–27. For this problem, Colton and Monk 24 considered the following acoustic scattering Equation (1.4)(4) where Ω_b is a ball of radius b in R³, c(x) > 0 (x ∈ R³) is the local speed of sound and satisfies c(x) = c₀ > 0 for r = |x| ≥ r₀, where c₀ is a constant and b ≥ r₀. Let n(x) = (c₀/c(x))² and k² = (ω/c₀)², where ω is the frequency of the incident wave, α is the direction of the incident wave, u^s(x) is the scattered field and is called as the Sommerfeld radiation condition. u^s(x) has the following asymptotic formulawhere are the spherical coordinates on the unit sphere ∂Ω in R³, is called as the scattering amplitude. Under some conditions, Equations (1.4) have a classical solution (v, w) 23,24. The following homogeneous equations (1.5) were discussed in 23,24,27,(5) It is well known that when n(x) = n(r)(r = |x|) radially symmetric, the solution to Equations (1.5) has the formwhere y_l is a solution ofwith initial conditionwhere the are spherical harmonics and the j_l are Bessel functions.

If the solutions to Equations (1.5) are only radially symmetric, then l = m = 0 andBy virtue of Equations (1.5), this yields(6) (7) (8) Let , z(ξ) = (n(r))^1/4y₀(r), from (1.6) and (1.7), we get(9) with the boundary condition at 0(10) where(11) Let , in virtue of (1.8) and note that n′(b) = 0, n(b) = 1 and , we obtain(12) Denoting , λ = B²k², a = b/B, by virtue of (1.9), (1.10) and (1.12), we get the Sturm–Liouville operator L(q, a).

Consider the following Sturm–Liouville operator defined by(13) with boundary conditions (1.2) and(14) where 0 ≤ ã < 1 or ã > 1, is a real-valued function and .

Mclaughlin and Polyakov 27 established the following remarkable uniqueness theorem for the original acoustic scattering problem, which is a generalization of the Hochstadt-Lieberman theorem (see 28,29).

Theorem 1.1 27, Theorem 1]

Let be real-valued functions in L²[0, 1]. λ_n(q) (n = 1, 2, …) be the spectrum of the Sturm–Liouville operator L(q, a) and be the spectrum of the Sturm–Liouville operator and . Ifandthen .

The following Lemma 1.2 is useful to prove main results.

Lemma 1.2 27, Lemma 2]

Let λ_n(n = 1, 2, …) be eigenvalues for the Sturm–Liouville operator L(q, a), then λ_n is a root of Equation (1.3) and there exists a positive integer i₀ such that λ_n is real for n > i₀ and satisfies the following asymptotic estimates:(15) where i₀ is integer part of , A = max{e^‖q‖, (‖q‖ + 1)e^‖q‖ + 1} and C is a constant.

This article is organized as follows. In Section 2, we present main results. In Section 3, the main results for the Sturm–Liouville operator L(q, a) will be proved.

2. Main results

Let s(x, λ) be solution of Equation (1.1) satisfying s(0, λ) = 0, s′(0, λ) = 1, and s(x, λ_n) be the eigenfunction corresponding to the eigenvalue λ_n of the Sturm–Liouville operator L(q, a). Let be the nodal points of the eigenfunction s(x, λ_n), i.e., , where . Denote the nodal interval by and the nodal length of the interval by . Let be the set of nodal points of the Sturm–Liouville operator L(q, a), where j = j(n) may be . For convenience, we still write j instead of j(n) (n sufficiently large).

First, we present asymptotic formulae for eigenvalues λ_n and zeros of the eigenfunction s(x, λ_n) of the Sturm–Liouville operator L(q, a) (n sufficiently large), respectively.

Lemma 2.1

For sufficiently large n, qualitative behaviour of eigenvalues for the Sturm–Liouville operator L(q, a) is as follows:

1.	In the case 0 ≤ a < 1, then(16)
2.	In the case a > 1, then(17)

Lemma 2.2

For sufficiently large n, the eigenfunction s(x, λ_n) of the Sturm–Liouville operator L(q, a) has at least nodal points in (0, 1) and

1.	In the case 0 ≤ a < 1,(18)
2.	In the case a > 1,(19) and(20) uniformly with respect to j ∈ Z+, where [x] denotes the largest integer ≤ x.

By virtue of (2.5) in Lemma 2.2, this yields the following corollary:

Corollary 2.3

The set is dense in [0, 1].

Next, the following inverse nodal problem is put forward.

Inverse problem Given a nodal points set or its subset X₀ ⊆ X, which is dense in (0, 1), we can find the potential q(x), coefficient a of the boundary condition and the spherically symmetric speed of sound n(r).

Now, main results are presented as follows.

Theorem 2.4

For each x ∈ (0, 1), let and satisfy , Then (2.6–2.10) hold.(21)

1.	In the case 0 ≤ a < 1,(22) and(23)
2.	In the case a > 1,(24) and(25)

According to Theorem 2.4, from the nodal data, we reconstruct the potential q(x) and coefficient a of the boundary condition for the Sturm–Liouville operator L(q, a). We provide a constructive procedure for the solution of the inverse nodal problem for the Sturm–Liouville operator L(q, a).

Theorem 2.5

Let be a subset of the nodal points set X and dense in (0, 1). From the nodal data X₀, we can construct the potential q(x) and coefficient a of the boundary condition by the following algorithm:

1.	For each x ∈ (0, 1), we choose a sequence such that .
2.	We calculate(26) i. In the case A > 1, we get a = 1 + A. ii. In the case 0 < A ≤ 1, we obtain a = 1 − A or a = 1 + A. iii. By virtue of (2.8) and (2.10), we calculate f(x) and g(x) in turn, respectively. iv. By taking derivatives for (2.8) and (2.10), respectively, we find that the potential q(x) satisfies(27) and(28)

Remark 2.6

When a = 0, the boundary condition (1.3) is z(1, λ) = 0, which do not depend on the spectral parameter λ, Theorem 2.5 was proved by McLaughlin 1, when a < 0, Theorem 2.5 is also valid.

By using only the nodal data in the reconstruction algorithm, we can reconstruct the potential q(x) and coefficient a of the boundary condition. Therefore, Theorem 2.7, i.e., a uniqueness theorem on the potential q(x) and coefficient a of the boundary condition obviously holds. Therefore, the proof of Theorem 2.7 is omitted here.

Theorem 2.7

Let be a subset of the nodal set X of the Sturm–Liouville operator L(q, a), which is dense in (0, 1) and be a subset of the nodal set of the Sturm–Liouville operator , which is dense in (0, 1). If , then(29) and

From Ref. 27, x = ξ/B, , , λ = B²k², a = b/B, , r = r(x)(0 ≤ x ≤ 1) and(30) (31) where b is a constant.

According to Theorem 2.7, by using the nodal points as data, we can also establish the following theorem for reconstruction of the spherically symmetric speed of sound n(r).

Theorem 2.8

Let be a subset of the nodal points set X and dense in (0, 1). Let n(r) ∈ C¹(R) be positive function, n″(r) ∈ L²[0, b] and n(b) = 1, n′(b) = 0, . If 0 < b < B and 0 < B < b are given, then the specification of X₀ is sufficient to determine the spherically symmetric speed of sound n(r) and n(r) is a solution of Equations (2.15) and (2.16). We can reconstruct the spherically symmetric speed of sound n(r) by the following algorithm:

1.	According to Theorem 2.7, in the case 0 ≤ a < 1 or a > 1, we reconstruct the potential q(x)(x ∈ (0, 1)) and coefficient a of the boundary condition, respectively.
2.	Solving Equations (2.15) and (2.16), we find the spherically symmetric speed of sound n(r) for the acoustic scattering problem.

3. Proofs

In this section, we prove the main results, respectively. First, we show that Lemma 2.1 and Lemma 2.2, respectively, hold, which play an important role in the proofs of the main theorems. Then, the proofs of Theorems 2.4 and 2.5 are presented.

Proof of Lemma 2.1

Let , from Lemma 1.2.1 in Ref. 28, p.6] and Lemma 1.2, the solution to Equation (1.1) satisfying s(0, λ) = 0, s′(0, λ) = 1 isandTherefore, we obtainandBy substituting the expressions of s(x, λ) and s′(x, λ) intowe get(32)

1.	In the case 0 ≤ a < 1, in virtue of (3.1), this yields(33)
2.	In the case a > 1, by virtue of (3.1), this yields(34) Thus, this completes the proof of Lemma 2.1.

Next, we prove that the asymptotic formulae (2.3) and (2.4) for the nodal points of eigenfunctions for the Sturm–Liouville operator L(q, a) hold.

Proof of Lemma 2.2

According to 30 and 31 the eigenfunction s(x, λ_n) has the following asymptotic formula for n → ∞ in x:Let s(x, λ_n) = 0, we obtain that the nodal points of the eigenfunction s(x, λ_n) satisfyBy using Taylor's expansions, we get the following formula for the nodal points as n → ∞ uniformly in j ∈ Z⁺:This yields(35) By virtue of (2.1), we have the following asymptotic formulae:

1.	In the case 0 ≤ a < 1, then(36) and(37) Substituting (3.5) and (3.6) into (3.4), we obtainHence, the formula (2.3) is valid.
2.	In the case a > 1, similarly we can show that the formula (2.4) holds. By virtue of (2.3) and (2.4), when 0 ≤ a < 1 or a > 1, for n sufficiently large, we get(38) uniformly with respect to j.

From (2.3) and (2.4), for , we have(39) In virtue of (3.8), for sufficiently large n, we can see that the eigenfunction s(x, λ_n) has at least zeros in the interval (0, 1).

Therefore, the proof of Lemma 2.2 is complete.

Now, we show that Theorem 2.4 and Theorem 2.5 hold and omit the proofs of Theorem 2.7 and Theorem 2.8.

Proof of Theorem 2.4

By virtue of (3.7), we have(40)

i.	In the case A > 1, we get a = 1 + A.
ii.	In the case 0 < A ≤ 1, we obtain a = 1 − A or a = 1 + A.

Letting , when 0 ≤ a < 1, from (2.3), we haveor when a > 1, from (2.4), we obtainwhere j = j(n) may be .

1.	In the case 0 ≤ a < 1, by using the Riemann–Lebesgue lemma and Lemma 2.2(1), we get(41)
2.	In the case a > 1, by using the Riemann–Lebesgue lemma and Lemma 2.2(2), we obtain(42) Thus, the proof of Theorem 2.4 is now complete.

Next, from Theorem 2.4, we prove Theorem 2.5.

Proof of Theorem 2.5

For a given nodal subset X₀, by virtue of Theorem 2.4, we can build up the reconstruction formula. From (2.6–2.10), we get the following procedure.

•	Step 1. Calculating , if A > 1, then a = 1 + A; if 0 < A ≤ 1, then a = 1 − A or a = 1 + A.
•	Step 2. For each x ∈ (0, 1), we choose a sequence such that .
•	Step 3. By virtue of (2.8) and (2.10), we calculate f(x) and g(x) in turn, respectively.
•	Step 4. By taking derivatives for (2.8) and (2.10), respectively, we get the potential q(x) such that(43) and(44) Thus, we can reconstruct the potential q(x) and coefficient a of the boundary condition.

Thus, this completes the proof of Theorem 2.5.

Finally, we present an example for the inverse nodal problem. i.e., given a nodal points set or its subset X₀ ⊆ X, which is dense in (0, 1), we can find the potential q(x) and coefficient a of the boundary condition.

Example

Let be a dense nodal set of the Sturm–Liouville operator L(q, a). Find the potential q(x) and coefficient a of the boundary condition.

By virtue of (3.7) and the assumption of the Example, we have(45) In virtue of (3.9) and (3.14), this yields(46) Therefore, or . When , from (3.10), we obtain(47) Hence(48) Substituting (3.17) into (3.12), we get(49) Integrating from 0 to 1 in (3.18), we have(50) Therefore(51) Similarly, when , we obtain q(x) = 2x.

Acknowledgements

The authors acknowledge helpful comments and valuable suggestions from the referees, which obviously have improved the quality of the original manuscript. This work was supported by the National Natural Science Foundation of China (No. 11171152) and the Natural Science Foundation of Jiangsu (No. BK 2010489).