On Inverse Nodal Problem and Multiplicities of Eigenvalues of a Vectorial Sturm-Liouville Problem

An m-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence fynj ,rðx, λ 2 nj ,rÞg ∞ j=1 of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix QðxÞ and A are simultaneously diagonalizable by the same unitary matrix U . Subsequently, some multiplicity results of eigenvalues are obtained.


Introduction
Consider the following m-dimensional vectorial Sturm-Liouville problem with the eigenparameter-dependent boundary conditions: −y″ + Q x ð Þy = λ 2 y, 0 ≤ x ≤ π, y 0 ð Þ = 0, Ay′ π ð Þ + λy π ð Þ = 0, where λ is the spectral parameter and y = ðy 1 , y 2 , ⋯, y m Þ T is an m-dimensional vectorial function. The potential matrix QðxÞ is an m × m real symmetric and nonnegative definite matrix-valued function which is defined and integrable on the interval ½0, π. A is an m × m nonsingular real symmetric matrix. Throughout this paper, we set m ≥ 2 and agree that 0 denotes the m-dimensional zero vector.
Sturm-Liouville problems with eigenparameter in the boundary conditions arise upon separation of variables in the one-dimensional wave and heat equations for various physical applications [1]. There are some literatures on such scalar problems (see [2][3][4][5]) and vectorial problems (see [6][7][8]). We consider the inverse nodal problem of (1) firstly. It was McLaughlin who initiated the study of the inverse nodal problem [9]. In the recent years, the inverse nodal problem of the Sturm-Liouville problem has been investigated a lot (see the monographs [10][11][12][13][14][15][16][17] and the references therein) and the vectorial inverse nodal problem is studied in [18,19]. However, McLaughlin's uniqueness theorem does not hold for the vectorial Sturm-Liouville problems. To clarify our problem, the following definitions could be seen in [18,19]. For the convenience of reader, we state here again. Let yðxÞ be an m-dimensional vectorial function defined on the interval ½0, π, if yðx 0 Þ = 0, x 0 ∈ ½0, π,x 0 will be called a nodal point (zero) of yðxÞ: We say that yðxÞ is a vectorial function of type (CZ) (or yðxÞ has a common zero property) if all the isolated zeros of its components are nodal points. The matrix QðxÞ is called simultaneously diagonalizable if there is a constant unitary matrix U such that U * QðxÞU is a diagonal matrixvalued function.
In 1999, Shen and Shieh [18] studied the inverse nodal problem of the vectorial equation in (1) with Dirichlet boundary condition when dimension m = 2: They proved that if the problem has infinitely many eigenfunctions fy n j ðxÞg ∞ j=1 which are of type (CZ), then QðxÞ is simultaneously diagonalizable. It seems to be the first study of the vectorial inverse nodal problem. Cheng et al. generalized to the arbitrary separated boundary conditions in [19]. Chan [6] investigated some eigenvalue problems of (1). He proved that the eigenvlaues of the problem are all real. Based on the results of [6,18,19], in this study, using a different method from that in [18,19], we consider the inverse nodal problem of (1). Firstly, we derived that the matrix Ð x 0 QðtÞdt is simultaneously diagonalizable which is equivalent to QðxÞ possessing the same property. That is, there exists a constant unitary matrix U such that U * QðxÞU = diag ðμ 1 ðxÞ, μ 2 ðxÞ, ⋯, μ m ðxÞÞ, where μ i ðxÞði = 1, 2, ⋯, mÞ are eigenvalues of QðxÞ: Meanwhile, if QðxÞ and A are simultaneously diagonalizable by the same unitary matrix U, we prove that the eigenfunctions of problem (1) are all of type (CZ) (see Corollary 3.2).
Next, we investigate the relationship between multiplicities of eigenvalues of problem (1) and matrix A. It is known that the eigenvalues of a scalar Sturm-Liouville problem with the separate boundary conditions are all simple. However, the multiplicities of the m-dimensional vectorial Sturm-Liouville problem may be among 1 to m. Such problems were studied in [20][21][22]. In [20], Shen and Shieh study the multiplicities of 2-dimensional vectorial Sturm-Liouville problem defined in [0, 1]. Suppose that QðxÞ = ðq ij ðxÞÞ is a continuous 2 × 2 Jacobian matrix-valued function; it is proved that if Ð 1 0 q 12 ðxÞdx ≠ 0, then the sufficiently large eigenvalues are simple. Subsequently, Kong [21] developed and improved the results in [20] to the case when QðxÞ is real symmetric. In 2007, Yang et al. [22] extended the result of [20,21] to the Sturm-Liouville equation with a weighted function w, a leading coefficient function p, and general separated conditions. From the estimation of eigenvalues (see Lemma 4), it is not difficult to see that the multiplicities of eigenvalues of problem (1) are determined firstly by the multiplicities of eigenvalues of matrix A no matter QðxÞ possesses any properties. We provide conditions on A and QðxÞ under which, with finitely many exceptions, the eigenvalues of vectorial problem (1) are simple. This paper is divided into four sections. Following this Introduction, in Section 2, we investigate the asymptotic expression of the eigenvalues and state some other preliminary lemmas for the main theorems. In Section 3, we discuss the inverse nodal problem of (1). In Section 4, some results about multiplicities of eigenvalues of the 2-dimensional vectorial Sturm-Liouville problem (1) are given.

Preliminaries
Let Yðx, λ 2 Þ be the matrix solutions of equation satisfying the initial conditions where E m denotes the m × m identity matrix and 0 m denotes the m × m zero matrix. Denote ΔðλÞ will be called the characteristic function of the eigenvalues of problem (1). The algebraic multiplicity of an eigenvalue λ is the order of λ as a zero of ΔðλÞ. The geometric multiplicity of λ as an eigenvalue of the problem (1) is defined to be the number of linearly independent solutions of the boundary problem. Let CðλÞ = AY ′ ðπ, λ 2 Þ + λYðπ, λ 2 Þ: Lemma 1. The geometric multiplicity of λ as an eigenvalue of problem (1) is equal to m − RankðCðλÞÞ.
If λ n,r is k-multiple, substituting λ n,r into (14) and noting that τ = 0, we get that ðε n,r Þ k = Oð1/nÞ: Consequently, ε n,r = Oð1/n 1/k Þ: Using the asymptotic expression of eigenvalues, we get that the asymptotic expression about the nodal points.

Lemma 5.
Suppose that y n,r ðx, λ 2 n,r Þ is an eigenfunction of the vectorial problem (1) of type ðCZÞ corresponding to the eigenvalue λ n,r , and fx n,r k g is the nodal set of y n,r ðx, λ 2 n,r Þ: Then, (i) for sufficiently large jnj,y n,r ðx, λ 2 n,r Þ possesses jnj nodal points x n,r k in ð0, πÞ (ii) the nodal points have the following asymptotic expression: for sufficiently large jnj Proof. The proof is similar to that of Lemma 2 in [18] and Theorem 2.3 in [19].

Inverse Nodal Problem
In this section, we discuss the inverse nodal problem of (1).

Theorem 6.
Suppose that there exists a sequence fy n j ,r ðx, λ 2 n j ,r Þg Proof. Denote v n j ,r a unit null vector of Cðλ n j ,r Þ, where Y ðx, λ 2 n j ,r Þ is the solution of matrix initial problem (2)-(3) corresponding to λ n j ,r . Then, is an eigenfunction of vectorial problem (1) corresponding to λ n j ,r . Assuming that y n j ,r ðx, λ 2 n j ,r Þ is a vectorial eigenfunction of type (CZ), by Lemma 5, we know that, y n j ,r ðx, λ 2 n j ,r Þ has jn j j nodal points x n j ,r k in ð0, πÞ: What is more, is dense in ½0, π; thus for a fixed x 0 in ð0, π, there exists a sequence of nodal x n j ,r k j such that lim j→∞ x n j ,r Since y n j ,r x n j ,r by (9), we have Y x n j ,r k j , λ 2 n j ,r = sin λ n j ,r x n j ,r k j Thus, ð x n j ,r k j 0 Q t ð Þdt · v n j ,r = 2λ n j ,r sin λ n j ,r x n j ,r k j cos λ n j ,r x n j ,r k j E m · v n j ,r + o 1 ð Þ · v n j ,r : Since kv n j ,r k = 1, v n j ,r converges to some unit vector denoting as v. Therefore, taking the limit as j ⟶ ∞ for both sides of Equation (24), we obtain where c 1 ðx 0 Þ is a scalar associated with x 0 . Since x 0 is arbitrary in ð0, π, we can choose x 1 ≠ x 0 in ð0, π and use the above argument for x 1 such that v is also an eigenvector of Ð x 1 0 QðtÞdt. Then, Ð x 0 0 QðtÞdt and Ð x 1 0 QðtÞdt have the same eigenvector v: That is, v is chosen independently of x: Furthermore, using Schmidt orthogonalization, v could expand into standard orthogonal basis denoting as u 1 , u 2 , ⋯, u m . Since Ð x 0 QðtÞdt is diagonalizable for all x ∈ ð0, π, u 2 , u 3 , ⋯, u m have to be eigenvectors for Ð x 0 QðtÞdt. Thus, there are other scalars c i ðxÞði = 2, 3, ⋯, mÞ depending on x such that Ð x 0 QðtÞdt · u i = c i ðxÞ · u i . Hence, we derive that Ð x 0 QðtÞdt is simultaneously diagonalizable. In fact, denote U = ½u 1 , u 2 , ⋯, u m , then we have Deriving the two sides of the above formula, we derived that matrix QðxÞ is also simultaneously diagonalizable by the same matrix U: That is, where μ i ðxÞ = c i ′ ðxÞði = 1, 2, ⋯, mÞ.
Since y n j ,r ðx, λ 2 n j ,r Þ satisfies the boundary conditions, hence, AY ′ π, λ 2 n j ,r v n j ,r + λ n j ,r Y π, λ 2 n j ,r v n j ,r = 0: ð28Þ By Lemma 2, we have Av n j ,r = −tan λ n j ,r π v n j ,r + O 1 n j ! : ð29Þ Let j ⟶ ∞, v n j ,r tends to the same vector v, and the sequence tan ðλ n j ,r πÞ has a limit −c ∈ ℝ: Thus, Av = cv: Subsequently, with a similar process, we conclude that A is diagonalizable by the same matrix U. That is, The proof of the theorem is finished. Proof.
(1) By the transformation y = Uz, the problem (1) becomes that is, Note that the two problems (1) and (32) have exactly the same eigenvalues and multiplicities. Let z = ðz 1 , z 2 , ⋯, z m Þ T . The vectorial problem (32) is equivalent to the following scalar Sturm-Liouville problems: All the eigenvalues of problem (32) are the collections of eigenvalues of scalar problems in (33). If λ ðiÞ * is an eigenvalue of the ith problem in (33) and z * ,i ðx, λ ðiÞ * Þ is the eigenfunction corresponding to λ ðiÞ * , then is the eigenfunction of the original problem (1) corresponding to λ ðiÞ * . Obviously, y * ðx, λ ðiÞ * Þ is a vectorial function of type (CZ).

Multiplicities of Eigenvalues
In this section, we first discuss the relationship between the multiplicities of eigenvalues of 2-dimensional vectorial Sturm-Liouville problem (1) and the matrix A in the boundary condition.
Lemma 8. When m = 2, the algebraic multiplicity and geometric multiplicity of λ as an eigenvalue of the problem (1) is equal.
The next theorem shows the relationship between multiplicities of eigenvalues and QðxÞ when m = 2 and A = cE 2 ðc ≠ 0Þ.
Theorem 10. Suppose that A = cE 2 ðc ≠ 0Þ: If the eigenvalues of the matrix Ð π 0 QðtÞdt have no repeated eigenvalues, then, with finitely many exceptions, the eigenvalues of problem (1) are all simple.
Proof. The proof of this conclusion is analogous to that of Theorem 9 and therefore is omitted. Note that formulas (9) and (10) are needed to calculate CðλÞ in this case.

Data Availability
No data were used to support the study.