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

Abstract

An -dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix and are simultaneously diagonalizable by the same unitary matrix . Subsequently, some multiplicity results of eigenvalues are obtained.

1. Introduction

Consider the following -dimensional vectorial Sturm-Liouville problem with the eigenparameter-dependent boundary conditions: where is the spectral parameter and is an -dimensional vectorial function. The potential matrix is an real symmetric and nonnegative definite matrix-valued function which is defined and integrable on the interval . is an nonsingular real symmetric matrix. Throughout this paper, we set and agree that denotes the -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 [2–5]) and vectorial problems (see [6–8]). 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 [10–17] 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 be an -dimensional vectorial function defined on the interval , if will be called a nodal point (zero) of We say that is a vectorial function of type (CZ) (or has a common zero property) if all the isolated zeros of its components are nodal points. The matrix is called simultaneously diagonalizable if there is a constant unitary matrix such that is a diagonal matrix-valued function.

In 1999, Shen and Shieh [18] studied the inverse nodal problem of the vectorial equation in (1) with Dirichlet boundary condition when dimension They proved that if the problem has infinitely many eigenfunctions which are of type (CZ), then 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 is simultaneously diagonalizable which is equivalent to possessing the same property. That is, there exists a constant unitary matrix such that where are eigenvalues of Meanwhile, if and are simultaneously diagonalizable by the same unitary matrix , 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 . 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 -dimensional vectorial Sturm-Liouville problem may be among 1 to . Such problems were studied in [20–22]. In [20], Shen and Shieh study the multiplicities of 2-dimensional vectorial Sturm-Liouville problem defined in [0, 1]. Suppose that is a continuous 2 Jacobian matrix-valued function; it is proved that if then the sufficiently large eigenvalues are simple. Subsequently, Kong [21] developed and improved the results in [20] to the case when is real symmetric. In 2007, Yang et al. [22] extended the result of [20, 21] to the Sturm-Liouville equation with a weighted function , a leading coefficient function , 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 no matter possesses any properties. We provide conditions on and 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.

2. Preliminaries

Let be the matrix solutions of equation satisfying the initial conditions where denotes the identity matrix and denotes the zero matrix. Denote

will be called the characteristic function of the eigenvalues of problem (1). The algebraic multiplicity of an eigenvalue is the order of as a zero of . The geometric multiplicity of as an eigenvalue of the problem (1) is defined to be the number of linearly independent solutions of the boundary problem. Let

Lemma 1. The geometric multiplicity of as an eigenvalue of problem (1) is equal to Rank.

From the above lemma, the geometric multiplicity of an eigenvalue of problem (1) is at most , and the geometric multiplicity of is if and only if is the zero matrix. It is known that is an entire function of order one with respect to What is more, when is nonnegative definite for any the eigenvalues of problem (1) are all real [6]. Denote By variation of constants, we have

Lemma 2 ([23]). For fixed the matrix functions and are entire functions with respect to variable When we have

Remark 3. Using the first relation in (6) to iterate equation (5), we find that By Riemann-Lebesgue lemma, Thus, when has the following asymptotic expansion: Similarly,

Lemma 4. Suppose that an eigenvalue is with algebraic multiplicity When , have the following asymptotic expansion: where are eigenvalues of matrix

Proof. Since is an nonsingular real symmetric matrix, there exists a unitary matrix such that where By formula (6) in Lemma 2, Thus, where and Additionally, Denote and Take a closed -curve On or for On similarly, we have for Denote On the contour since we have that is, Using Rouché theorem, the number of zeros of inside coincides with the number of zeros of ; i.e., it equals to (counting the multiple zeros). Note that the zeros of and are all real. Now, we consider the zeros of in They are Now, it is easy to see that the algebraic multiplicity of an eigenvalue of problem (1) is between to . Note that the eigenvalues of problem (1), in this case, stretch from to . Consequently, we denote all the eigenvalues as For sufficiently large , lies between and . Set . Moreover, denote where . Applying the Rouché theorem again to the circle if is -multiple in , there are zeros of For sufficiently small , it follows that .
If is -multiple, substituting into (14) and noting that , we get that Consequently,

Using the asymptotic expression of eigenvalues, we get that the asymptotic expression about the nodal points.

Lemma 5. Suppose that is an eigenfunction of the vectorial problem (1) of type corresponding to the eigenvalue , and is the nodal set of Then, (i)for sufficiently large possesses nodal points in (ii)the nodal points have the following asymptotic expression: for sufficiently large (iii) is dense in

Proof. The proof is similar to that of Lemma 2 in [18] and Theorem 2.3 in [19].

3. Inverse Nodal Problem

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

Theorem 6. Suppose that there exists a sequence of eigenfunctions of problem (1) which are all vectorial functions of type , then and are simultaneously diagonalizable by the same unitary matrix .

Proof. Denote a unit null vector of , where is the solution of matrix initial problem (2)–(3) corresponding to . Then, is an eigenfunction of vectorial problem (1) corresponding to . Assuming that is a vectorial eigenfunction of type (CZ), by Lemma 5, we know that, has nodal points in What is more, is dense in ; thus for a fixed in , there exists a sequence of nodal such that Since by (9), we have Thus, Since , converges to some unit vector denoting as . Therefore, taking the limit as for both sides of Equation (24), we obtain where is a scalar associated with . Since is arbitrary in , we can choose in and use the above argument for such that is also an eigenvector of . Then, and have the same eigenvector That is, is chosen independently of
Furthermore, using Schmidt orthogonalization, could expand into standard orthogonal basis denoting as . Since is diagonalizable for all , have to be eigenvectors for . Thus, there are other scalars depending on such that . Hence, we derive that is simultaneously diagonalizable. In fact, denote , then we have Deriving the two sides of the above formula, we derived that matrix is also simultaneously diagonalizable by the same matrix That is, where .
Since satisfies the boundary conditions, hence, By Lemma 2, we have Let , tends to the same vector , and the sequence has a limit Thus, Subsequently, with a similar process, we conclude that is diagonalizable by the same matrix . That is, The proof of the theorem is finished.

Theorem 7. Suppose that and are simultaneously diagonalizable by the same unitary matrix . Then, (1)the eigenfunctions of problem (1) are all of type (2)the algebraic multiplicities and geometric multiplicities of eigenvalues of problem (1) are equal

Proof. (1)By the transformation , the problem (1) becomes that is,  Note that the two problems (1) and (32) have exactly the same eigenvalues and multiplicities. Let . 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 is an eigenvalue of the th problem in (33) and is the eigenfunction corresponding to , then is the eigenfunction of the original problem (1) corresponding to . Obviously, is a vectorial function of type (CZ).(2)Let be the solutions of equations in (33) satisfying the initial conditions We have Thus, If is a -multiple (geometric) eigenvalue, then Rank() =  We may as well let the last elements in the diagonal of be zero. Hence, is an eigenvalue of the last problems in (33). Denote We have and for Thus, is also with algebraic multiplicity .

4. 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 in the boundary condition.

Lemma 8. When , the algebraic multiplicity and geometric multiplicity of as an eigenvalue of the problem (1) is equal.

Proof. See [6].

Theorem 9. Let . If the eigenvalues of matrix are all simple, then with finitely many exceptions, the eigenvalues of problem (1) are all simple too.

Proof. Let be an eigenvalue of problem (1). By Lemma 4, Suppose, to the contrary, that the multiplicities of the eigenvalues of problem (1) are 2. By Lemma 1, . That is, By (13), we have Calculating directly and noting that the eigenvalues are real, we get Taking the limit as , (39) follows that which contradicts with the facts that the eigenvalues of the matrix are all simple.
The next theorem shows the relationship between multiplicities of eigenvalues and when and .

Theorem 10. Suppose that If the eigenvalues of the matrix 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 in this case.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work is supported by the National Scientific Foundation of China (Grant Nos. 11801012 and 11771017) and the Inner Mongolia Natural Science Foundation (Grant No. 2017MS0125).