Direct and inverse nodal problem for differential pencil with coupled boundary conditions

Abstract

We consider the eigenvalue problem and inverse nodal problem for the s-wave Schrödinger equation with a radial static potential q(x) + 2λp(x) and with coupled boundary conditions on a finite segment. First, we establish an identity for the first regularized trace of this operator. Second, we prove that a dense subset of nodal points uniquely determines the parameters in the boundary conditions and the potential functions p(x) and q(x). We also provide a constructive procedure for the solution of the inverse nodal problem.

Keywords:

• differential pencil
• coupled boundary conditions
• eigenvalue problem
• regularized trace formula
• inverse nodal problem

1. Introduction

This article deals with the eigenvalue problem and inverse nodal problem for the s-wave Schrödinger equation 1 (1) on the interval (0, π) with coupled boundary conditions (2) where λ is the spectral parameter, and q ∈ L[0, π] are real-valued functions, 0 ≠ b is a real number and θ = 0, 1. The goal of this article is to calculate the first-order regularized trace for (1.1)–(1.2), and to reconstruct unknown parameters in (1.1)–(1.2) from known nodal set for eigenfunctions.

The theory of regularized traces of ordinary differential operators has a long history. First, the trace formulas for the Sturm–Liouville operator with the Dirichlet boundary conditions and sufficiently smooth potential were established in 2,3. Afterwards, these investigations were continued in many directions (see, 4–16, etc.). A method for calculating trace formulas for general problems involving ordinary differential equations on a finite interval was proposed in 17. The bibliography on the subject is very extensive and we refer to the list of the works in 14,18.

The trace formulas can be used for approximate calculation of the first eigenvalues of an operator 14,19, and in order to establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of an operator 20.

Inverse nodal problems consist in recovering operators from their nodal points (zeros of eigenfunctions). The inverse nodal problem was posed and solved for Sturm–Liouville problems by McLaughlin 21. The problems of this type are related to some questions of mechanics and mathematical physics 21. Inverse nodal problems for the Sturm–Liouville operators with separated boundary conditions on an interval are studied in sufficient detail in 21–33. Cheng and Law first studied the inverse nodal problem for Hill's equation, i.e. the Sturm–Liouville operators with coupled boundary conditions 34. Except for a bibliography 34, we have not seen another result on inverse nodal problems for the Sturm–Liouville operators with coupled boundary conditions.

We also point out that the trace formulas for the differential pencil (1.1) with separated boundary conditions, periodic or antiperiodic boundary conditions were obtained in 6,15,16, and the inverse nodal problem of the differential pencil with separated boundary conditions was studied in 26 (with an excessive assumption that the function p(x) was known a priori) and 35, in 36 for the case of the Robin boundary conditions, and in 37 for the Dirichlet boundary conditions. However, the trace and inverse nodal problem for differential pencil with coupled boundary conditions have never been considered before.

The main results of this article are Theorems 3.1 (trace formulas) and 5.2 (reconstruction formulae from nodes), where the uniqueness for the solution of the inverse nodal problem is proved and the reconstruction formulae for the coefficients of the pencil are provided. Note that the presence of irregular boundary conditions (1.2) makes the investigation of both the oscillation of eigenfunctions and the inverse nodal problem more difficult.

This article is organized as follows. Section 2 is devoted to the analysis of the characteristic determinant. In Section 3, a direct problem is considered, i.e. regularized trace formula of this operator. Section 4 is concerned with the oscillation of the eigenfunctions corresponding to large modulus eigenvalues and the asymptotic of the nodal points. Section 5 is devoted to the inverse nodal problems, i.e. a uniqueness theorem is proved and a solution algorithm for the recovery of p, q, θ, b is presented from their nodal data.

2. Analysis of the characteristic determinant

Denote by ψ(x, λ), ϕ(x, λ) the fundamental system of solutions to (1.1) with the initial conditions ψ(0, λ) = ϕ′(0, λ) = 1, ψ′(0, λ) = ϕ(0, λ) = 0. Simple calculations show that the characteristic equation of (1.1)–(1.2) can be reduced to the form Δ(λ) = 0, where (3)

To analyse the characteristic determinant Δ(λ), we need more precise asymptotic formulas for the functions ϕ(x, λ), ϕ′(x, λ) and ψ(x, λ). The results have been obtained, see Lemma 3.1 and Corollary 3.2 in 16.

Let ψ(x, λ) and ϕ(x, λ) be the solutions to equation (1.1) with the initial conditions then for each fixed x these solutions are entire functions of the parameter λ and the following representations hold 16: where τ = |ℑλ|, and It is easy to obtain asymptotic expression of the function ϕ′(x, λ) 16: where

For the convenience, we now set

Substituting into (2.1) the expressions for ϕ(π, λ), ϕ′(π, λ), ψ(π, λ) with ⟨p⟩ = 0, respectively, we obtain

Denote and

Define . Then , are zeroes of the function Δ₀(λ) and simple algebraically.

3. Direct problem I: characterization of spectrum

In this section, we will prove the following statement.

Theorem 3.1

Let q(x) ∈ L[0, π] and be an arbitrary real-valued functions and denote by λ_n the eigenvalues of (1.1)–(1.2). Then the sequence satisfies the asymptotic form (4) and we have the trace formula for (5) and for (6) where and .

First, we consider the case ⟨p⟩ = 0. Denote by C_n the circle of radius , centered at the origin , , and by Γ_N the counterclockwise square contour with four vertices where N is a natural number. Obviously, if λ ∈ C_n or λ ∈ Γ_N, then |Δ₀(λ)| ≥ Me^τπ/|λ| (M > 0) by using similar method in 38–40. Thus, on λ ∈ C_n or λ ∈ Γ_N, we have (7) Expanding by the Maclaurin formula, we find that (8) Using the residue calculation the following identities are true: and which can lead to (9) Thus, we have (10) Substituting into (3.7), the expressions for c₁, d₁, d₂, c₂, c₃, a₁ and b₁ (see Section 2 in page 5), we obtain (11) Thus, we have the asymptotic formula of the eigenvalues for the problem (1.1)–(1.2) with ⟨p⟩ = 0.

It is well known that the eigenvalues of (1.1)–(1.2) form a sequence , . This asymptotic relation for the eigenvalues implies that, for all sufficiently large N, the numbers λ_n with |n| ≤ N are inside Γ_N, and the numbers λ_n with |n| > N are outside Γ_N. It follows that (12)

Using the residue calculation we have the following identities: and First, we consider the case with . Using the above identities and equation (3.9), we get (13) That is, (14) Passing to the limit as N → ∞ in (3.11), we have (15) Using well-known formula we have Thus, we obtain (16) Substituting into (3.13) the expressions for c₁, d₁, d₂, c₂, c₃, a₁ and b₁ (see Section 2 in page 5), we obtain (17)

Next we consider the case with . Using Equation (3.9), we get (18) That is, (19) Passing to the limit as N → ∞ in (3.16), we have (20) Substituting into (3.17) the expressions for d₁, c₃, a₁ (see Section 2 in page 5), we obtain (21)

Now, we consider the case ⟨p⟩ ≠ 0. By a direct calculation we note that the equation is equivalent to Let and then at this case, we have and .

Suppose that (1.1)–(1.2) with potentials and has eigenvalues . According to (3.8), (3.14) and (3.18), we have for and for ,

Substituting the expressions for , and into these equalities, respectively, we obtain formulas (3.1), (3.2) and (3.3).

4. Direct problem II: oscillation and the nodal points

Now, we study the eigenfunction y(x, λ_n) corresponding to eigenvalue λ_n. Moreover, for large |n| the λ_n and y(x, λ_n) corresponding to eigenvalue λ_n are real. Using an analog of the Sturm oscillation theorem, for sufficiently large |n|, we find that y(x, λ_n) has exactly N(n, γ) nodal points in (0, π), where and Suppose are the nodal points of the eigenfunction y(x, λ_n). In other words, .

We know that the general solution of Equation (1.1) has the form where and are two arbitrary real number. If is an eigenfunction corresponding to eigenvalue λ_n for the problem (1.1) and (1.2), then the following linear system of equations about variables and possesses non-vanishing solutions It is easy to see that thus we may take . By substituting the expressions of ϕ(π, λ_n) and ψ(π, λ_n) into and , we have (22) Then, we have i.e. (23)

Denote (24)

Lemma 4.1

For sufficiently large |n|, the eigenfunction y(x, λ_n) of the problem (1.1) and (1.2) has exactly N(n, γ) nodes in (0, π). Moreover, as |n| → ∞, (25) uniformly with respect to j ∈ J(n, γ), where

Proof

According to (3.1), for large |n|, in the domain there is exactly one eigenvalue λ_n. Moreover, for large |n| the λ_n is real. By calculations, we get which implies that thus, Taking (4.2) into account and letting y(x, λ_n) = 0, we obtain (26) Note that and Thus, Equation (4.5) can be reduced to the form Moreover, we obtain (27)

Direct calculations imply that and Substituting these equations into (4.6) we get (28)

Denote (29) then Using trigonometrical calculations yields (30)

Using Taylor's expansion for the arctangent, we have that is, (31) From (3.1), we deduce (32) Substituting (4.11) into (4.10), it follows that that is, the asymptotic formula (4.4) for nodal points as |n| → ∞ uniformly in j ∈ Z:

The equality (4.4) gives uniformly with respect to j. Consequently, for large |n| we have for positive n and for negative n. For j = 0, 1, … , n (n > 0), the formula (4.4) gives Thus, according to the order of , for large |n|, the eigenfunction y(x, λ_n) has exactly N(n, γ) nodes in the interval (0, π), i.e. , for positive n.

For j = 0, −1, −2, … , n + 1, n, n − 1 (n < 0), the formula (4.4) gives The eigenfunction y(x, λ_n) has exactly N(n, γ) nodes in the interval (0, π), i.e. , for negative n. The proof is complete.

Remark 4.1

From Lemma 4.1, it follows that the set is dense in [0, π]. Therefore, for all x ∈ [0, π], there exists such that as |n| → ∞. Moreover, notice that

5. Inverse nodal problems: uniqueness theorem and algorithm

We see that there exists N₀ such that for all |n| > N₀ the eigenfunction y(x, λ_n) of the problem (1.1) and (1.2) has exactly N(n, γ) (simple) nodes in the interval (0, π), i.e. . The set is called the nodes of the problem (1.1) and (1.2).

We also define the function j_n(x) to be the largest index j such that for n > 0, and the function j_n(x) to be the largest index |j| such that for n < 0. Thus, j = j_n(x) if and only if for n > 0, and for n < 0.

We consider the following inverse problem.

Problem

Given a set X of nodal points or a subset of a set X, where is dense in (0, π), find the parameters θ, b in the boundary conditions, and the potentials p(x), q(x) in Equation (1.1).

Theorem 5.1

For each x ∈ [0, π]. Let be chosen such that  = x. Then the following finite limits exist and the corresponding equalities hold: (33) (34) and (35)

Proof

Using the asymptotic expansions (4.4) for nodal points, we get (36) and (37) Also, the fact that implies that and From this it follows that as |n| → ∞ the limits of left-hand sides in (5.4) and (5.5) exist, and equations (5.3) hold. This proves the theorem.

Let us now formulate a uniqueness theorem and provide a constructive procedure for the solution of the inverse nodal problem.

Denote and Selecting a nodal set X_odd or X_even, then denote g_o(x) by reconstructed from a nodal set X_odd, and g_e(x) by reconstructed from a nodal set X_even.

Theorem 5.2

Let and X⁰ ⊂ X be a subset of nodal points which satisfy that is dense in (0, π). Then the X⁰ uniquely determines the potential p(x) − ⟨p⟩ in (0, π), q(x) on (0, π), and the coefficients θ, b in the boundary conditions. The potentials p(x) − ⟨p⟩, q(x) and the numbers θ, b can be constructed via the following algorithm:

1.	each x ∈ [0, π] choose a sequence such that as |n| → ∞;
2.	find the function f(x) via (5.1) and in turn calculate (38) and when and
3.	calculate g(x) by formula (5.2) and in turn determine (39)

Proof

Now given a nodal subset X⁰, by Theorem 5.1, we can build up the reconstruction formulae.

Formulae (5.6) and (5.7) can be derived from (5.3) stepwise. We obtain the following procedure.

	Step 1 Take value for f(x) at x = 0, we have
	Step 2 Take value for f(x) at x = π, we have then it yields
	Step 3 Take derivatives of the function f(x), we obtain which implies that
	Step 4 When , using (4.8) and we can obtain and
	Step 5 Selecting a nodal set Xeven, to construct ge(x) from a nodal set Xeven.
	Step 6 Take values for ge(x) at x = 0, π, respectively, we get then it yields that for
	Step 7 After , A0, p(x) are reconstructed, from we obtain

Since each nodal data only determine a set of reconstruction formulae, the uniqueness holds obviously.

From reconstruction formula (5.6) in Theorem 5.2, we can judge whether the parameter or . The parameters θ and b have been constructed under condition . When , how to reconstruct the parameters θ and b?

Remark 5.1

When , taking values for g(x) at x = 0, π, respectively, we get then it yields that

I.

Selecting a nodal set Xodd, then (i1) For go(x) reconstructed from a nodal set Xodd, if go(0) + go(π) − π⟨p⟩2 ≠ 0, then θ = 1 and After reconstructing b1 and A0, returning to Step 6, we shall obtain q(x). (i2) For go(x) reconstructed from a nodal set Xodd, if go(0) + go(π) − π⟨p⟩2 = 0, (i2−1) Selecting ge(x) reconstructed from a nodal set Xeven, if ge(0) + ge(π) − π⟨p⟩2 = 0, then After reconstructing b1 = 0 and A0, returning to Step 6, we shall obtain q(x); (i2−2) selecting ge(x) reconstructed from a nodal set Xeven, if ge(0) + ge(π) − π⟨p⟩2 ≠ 0, then After reconstructing b1 = 0 and A0, returning to Step 6, we shall obtain q(x).

II.

Selecting a nodal set Xeven, then (ii1) For ge(x) reconstructed from a nodal set Xeven, if ge(0) + ge(π) − π⟨p⟩2 ≠ 0, then θ = 0 and After reconstructing b1 and A0, returning to Step 6 in the proof of Theorem 5.2, we shall obtain q(x). (ii2) For ge(x) reconstructed from a nodal set Xeven, if ge(0) + ge(π) − π⟨p⟩2 = 0, (ii2−1) selecting go(x) reconstructed from a nodal set Xodd, if go(0) + go(π) − π⟨p⟩2 = 0, then After reconstructing b1 = 0 and A0, returning to Step 6, we shall obtain q(x); (ii2−2) selecting go(x) reconstructed from a nodal set Xodd, if go(0) + go(π) − π⟨p⟩2 ≠ 0, then After reconstructing b1 = 0 and A0, returning to Step 6, we shall obtain q(x).

Acknowledgements

The author would like to thank the referees for valuable comments. The author is also grateful to Professor V.N. Pivovarchik and Professor V.A. Yurko for many useful discussions related to some topics of spectral analysis of differential operators. This work was supported by the National Natural Science Foundation of China (11171152/A010602), Natural Science Foundation of Jiangsu Province of China (BK 2010489) and the Outstanding Plan-Zijin Star Foundation of NUST (AB 41366).