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

Authors: Xiaoyun Liu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

Anm-dimensional vectorial inverse nodal Sturm-Liouville problem with eigenparameter-dependent boundary conditions is studied. We show that if there exists an infinite sequence ynj,rx,λnj,r2j=1∞ of eigenfunctions which are all vectorial functions of type (CZ), then the potential matrix Qx and A are simultaneously diagonalizable by the same unitary matrix U. Subsequently, some multiplicity results of eigenvalues are obtained.


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## Body

## 1. Introduction

Consider the followingm-dimensional vectorial Sturm-Liouville problem with the eigenparameter-dependent boundary conditions:
(1)−y″+Qxy=λ2y,0≤x≤π,y0=0,Ay′π+λyπ=0,where λ is the spectral parameter and y=y1,y2,⋯,ymT is an m-dimensional vectorial function. The potential matrix Qx is an m×m real symmetric and nonnegative definite matrix-valued function which is defined and integrable on the interval 0,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 [2–5]) 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 yx be an m-dimensional vectorial function defined on the interval 0,π, if yx0=0,x0∈0,π,x0 will be called a nodal point (zero) of yx. We say that yx is a vectorial function of type (CZ) (or yx has a common zero property) if all the isolated zeros of its components are nodal points. The matrix Qx is called simultaneously diagonalizable if there is a constant unitary matrix U such that U∗QxU 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 m=2. They proved that if the problem has infinitely many eigenfunctions ynjxj=1∞ which are of type (CZ), then Qx 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 ∫0xQtdt is simultaneously diagonalizable which is equivalent to Qx possessing the same property. That is, there exists a constant unitary matrix U such that U∗QxU=diagμ1x,μ2x,⋯,μmx, where μixi=1,2,⋯,m are eigenvalues of Qx. Meanwhile, if Qx 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 [20–22]. In [20], Shen and Shieh study the multiplicities of 2-dimensional vectorial Sturm-Liouville problem defined in [0, 1]. Suppose that Qx=qijx is a continuous 2×2 Jacobian matrix-valued function; it is proved that if ∫01q12xdx≠0, then the sufficiently large eigenvalues are simple. Subsequently, Kong [21] developed and improved the results in [20] to the case when Qx 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 Lemma4), 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 Qx possesses any properties. We provide conditions on A and Qx 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 Section2, 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

LetYx,λ2 be the matrix solutions of equation
(2)−Y″+QxY=λ2Y,x∈0,π,satisfying the initial conditions
(3)Y0,λ2=0m,Y′0,λ2=Em,where Em denotes the m×m identity matrix and 0m denotes the m×m zero matrix. Denote
(4)Δλ=detAY′π,λ2+λYπ,λ2.Δλ 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 Δλ. 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 Cλ=AY′π,λ2+λYπ,λ2.Lemma 1.
The geometric multiplicity ofλ as an eigenvalue of problem (1) is equal to m−RankCλ.From the above lemma, the geometric multiplicity of an eigenvalueλ∗ of problem (1) is at most m, and the geometric multiplicity of λ∗ is m if and only if Cλ∗ is the zero matrix. It is known that Δλ is an entire function of order one with respect to λ. What is more, when Qx is nonnegative definite for any x∈0,π, the eigenvalues of problem (1) are all real [6]. Denote λ=σ+iτ,σ,τ∈ℝ. By variation of constants, we have
(5)Yx,λ2=sinλxλEm+1λ∫0xsinλx−tQtYt,λ2dt.Lemma 2 ([23]).
For fixedx∈0,π, the matrix functions Yx,λ2 and Y′x,λ2 are entire functions with respect to variable λ. When λ⟶∞, we have
(6)Yx,λ2=sinλxλEm+Oeτxλ2,Y′x,λ2=cosλxEm+Oeτx∣λ∣.Remark 3.
Using the first relation in (6) to iterate equation (5), we find that
(7)Yx,λ2=sinλxλEm−cosλx2λ2∫0xQtdt+12λ2∫0xcosλx−2tQtdt+Oeτxλ3.
By Riemann-Lebesgue lemma,(8)limλ→∞∫0xcosλx−2tQtdt=0.
Thus, whenλ⟶∞,Yx,λ2 has the following asymptotic expansion:
(9)Yx,λ2=sinλxλEm−cosλx2λ2∫0xQtdt+oeτxλ2.
Similarly,(10)Y′x,λ2=cosλxEm+sinλx2λ∫0xQtdt+oeτxλ.Lemma 4.
Suppose that an eigenvalueλn,rr=1,2,⋯,m is with algebraic multiplicity k1≤k≤m. When n⟶∞, λn,r have the following asymptotic expansion:
(11)λn,r=n−1πtan−1ar+O1n1/k.where ar are eigenvalues of matrix A.Proof.
SinceA is an m×m nonsingular real symmetric matrix, there exists a unitary matrix U such that
(12)U∗AU=diaga1,a2,⋯,am,where a1≤a2≤⋯≤am. By formula (6) in Lemma 2,
(13)U∗CλU=diaga1,a2,⋯,amcosλπ+sinλπEm+Oe∣τ∣π∣λ∣.
Thus,(14)Δλ=detCλ=detU∗CλU=Πr=1marcosλπ+sinλπ+Oemτπλ=Πr=1mRrsinλπ+θr+Oemτπλ,where Rr=ar2+11/2,θr=tan−1ar and θr<π/2. Additionally, θ1≤θ2≤⋯≤θm. Denote Δ0λ=Πr=1mRrsinλπ+θr and δλ=Oem∣τ∣π/λ. Take a closed λ-curve Γn=Γn1∪Γn2∪Γn3n=1,2,⋯:(15)Γn1=λ=σ+iτ∣σ=n+12−θ1π,0≤τ≤n,Γn2=λ=σ+iτ∣σ=−n+12−θmπ,0≤τ≤n,Γn3=λ=σ+iτ∣−n+12−θmπ≤σ≤n+12−θ1π,τ=n.
OnΓn1 or Γn2, for r=1,2,⋯,m,(16)sinλπ+θr=12ieiσπ+θre−τπ−e−iσπ+θreτπ≥eτπ21−e2iσπ+θre−2τπ>eτπ4.
OnΓn3, similarly, we have sinλπ+θr>e∣τ∣π/4 for r=1,2,⋯,m. Denote M=Πr=1mRr. On the contour Γn, since λ≥1, we have
(17)Δ0λΓn=Πr=1mRrsinλπ+θrΓn>M·emτπ4m·λ,that is, Δ0λΓn>δλΓn. Using Rouché theorem, the number of zeros of Δλ inside Γn coincides with the number of zeros of Δ0λ; i.e., it equals to 2n+1m (counting the multiple zeros). Note that the zeros of Δλ and Δ0λ are all real. Now, we consider the zeros of Δ0λ in Γn. They are
(18)λj,r0=j−θrπ,j=0,±1,⋯,±n,r=1,2,⋯,m.
Now, it is easy to see that the algebraic multiplicity of an eigenvalueλ of problem (1) is between 1 to m. Note that the eigenvalues of problem (1), in this case, stretch from −∞ to +∞. Consequently, we denote all the eigenvalues as λn,rn=0,±1,±2,⋯;r=1,2,⋯,m. For sufficiently large n, λn,r lies between Γn−1 and Γn. Set λn,r=λn,r0+εn,r=n−θr/π+εn,r. Moreover, denote Cδn,r≔λ:λ−λn,r0≤δ, where 0<δ<min1≤i,j≤m,i≠j1/πθi−θj. Applying the Rouché theorem again to the circle Cδn,r, if λn,r0 is k-multiple 1≤k≤m, in Cδn,r, there are k zeros of Δλ. For sufficiently small δ, it follows that εn,r=o1.
Ifλn,r is k-multiple, substituting λn,r into (14) and noting that τ=0, we get that εn,rk=O1/n. Consequently, εn,r=O1/n1/k.Using the asymptotic expression of eigenvalues, we get that the asymptotic expression about the nodal points.Lemma 5.
Suppose thatyn,rx,λn,r2 is an eigenfunction of the vectorial problem (1) of type CZ corresponding to the eigenvalue λn,r, and xkn,r is the nodal set of yn,rx,λn,r2. Then,
(i)
for sufficiently largen,yn,rx,λn,r2 possesses n nodal points xkn,r in 0,π(ii)
the nodal points have the following asymptotic expression:(19)xkn,r=k+1πn+O1n2,k=0,1,⋯,n−1,for sufficiently large n(iii)
xkn,r:0≤k≤n−1n=1∞ is dense in 0,πProof.
The proof is similar to that of Lemma2 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 sequenceynj,rx,λnj,r2j=1∞ of eigenfunctions of problem (1) which are all vectorial functions of type CZ, then Qx and A are simultaneously diagonalizable by the same unitary matrix U.Proof.
Denotevnj,r a unit null vector of Cλnj,r, where Yx,λnj,r2 is the solution of matrix initial problem (2)–(3) corresponding to λnj,r. Then,
(20)ynj,rx,λnj,r2=Yx,λnj,r2vnj,ris an eigenfunction of vectorial problem (1) corresponding to λnj,r. Assuming that ynj,rx,λnj,r2 is a vectorial eigenfunction of type (CZ), by Lemma 5, we know that, ynj,rx,λnj,r2 has nj nodal points xknj,r in 0,π. What is more, xknj,r∣k=0,1,⋯,nj−1j=1∞ is dense in 0,π; thus for a fixed x0 in 0,π, there exists a sequence of nodal xkjnj,r such that
(21)limj→∞xkjnj,r=x0.
Since(22)ynj,rxkjnj,r,λnj,r2=Yxkjnj,r,λnj,r2vnj,r=0,by (9), we have
(23)Yxkjnj,r,λnj,r2=sinλnj,rxkjnj,rλnj,rEm−cosλnj,rxkjnj,r2λnj,r2∫0xkjnj,rQtdt+o1λnj,r2.
Thus,(24)∫0xkjnj,rQtdt·vnj,r=2λnj,rsinλnj,rxkjnj,rcosλnj,rxkjnj,rEm·vnj,r+o1·vnj,r.
Sincevnj,r=1, vnj,r converges to some unit vector denoting as v. Therefore, taking the limit as j⟶∞ for both sides of Equation (24), we obtain
(25)∫0x0Qtdt·v=c1x0·v,where c1x0 is a scalar associated with x0. Since x0 is arbitrary in 0,π, we can choose x1≠x0 in 0,π and use the above argument for x1 such that v is also an eigenvector of ∫0x1Qtdt. Then, ∫0x0Qtdt and ∫0x1Qtdt 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 u1,u2,⋯,um. Since ∫0xQtdt is diagonalizable for all x∈0,π, u2,u3,⋯,um have to be eigenvectors for ∫0xQtdt. Thus, there are other scalars cixi=2,3,⋯,m depending on x such that ∫0xQtdt·ui=cix·ui. Hence, we derive that ∫0xQtdt is simultaneously diagonalizable. In fact, denote U=u1,u2,⋯,um, then we have
(26)U∗∫0xQtdtU=diagc1x,c2x,⋯,cmx.
Deriving the two sides of the above formula, we derived that matrixQx is also simultaneously diagonalizable by the same matrix U. That is,
(27)U∗QxU=Λx=diagμ1x,μ2x,⋯,μmx,where μix=ci′xi=1,2,⋯,m.
Sinceynj,rx,λnj,r2 satisfies the boundary conditions, hence,
(28)AY′π,λnj,r2vnj,r+λnj,rYπ,λnj,r2vnj,r=0.
By Lemma2, we have
(29)Avnj,r=−tanλnj,rπvnj,r+O1nj.
Letj⟶∞, vnj,r tends to the same vector v, and the sequence tanλnj,rπ 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,
(30)U∗AU=ΛA=diaga1,a2,⋯,am.
The proof of the theorem is finished.Theorem 7.
Suppose thatQx and A are simultaneously diagonalizable by the same unitary matrix U. Then,
(1)
the eigenfunctions of problem (1) are all of type CZ(2)
the algebraic multiplicities and geometric multiplicities of eigenvalues of problem (1) are equalProof.
(1)
By the transformationy=Uz, the problem (1) becomes
(31)z″=U−1y″=U−1QUz−λ2Uz,z0=0,ΛAz′π+λzπ=0,that is,
(32)−z″+Λxz=λ2z,z0=0,ΛAz′π+λzπ=0.Note that the two problems (1) and (32) have exactly the same eigenvalues and multiplicities. Let z=z1,z2,⋯,zmT. The vectorial problem (32) is equivalent to the following scalar Sturm-Liouville problems:
(33)−zi′′+μixzi=λ2zi,x∈0,π,zi0=0,aizi′π+λziπ=0,i=1,2,⋯,m.All the eigenvalues of problem (32) are the collections of eigenvalues of scalar problems in (33). If λ∗i is an eigenvalue of the ith problem in (33) and z∗,ix,λ∗i is the eigenfunction corresponding to λ∗i, then
(34)y∗x,λ∗i=U0,⋯,z∗,ix,λ∗i,⋯,0Tis the eigenfunction of the original problem (1) corresponding to λ∗i. Obviously, y∗x,λ∗i is a vectorial function of type (CZ).(2)
Letz0,ix,λ2i=1,2,⋯,m be the solutions of equations in (33) satisfying the initial conditions
(35)z0,i0,λ2=0,z0,i′0,λ2=1
We have(36)Cλ=diaga1z0,1′π,λ2+λz0,1π,λ2,a2z0,2′π,λ2+λz0,2π,λ2,⋯,amz0,m′π,λ2+λz0,mπ,λ2.
Thus,Δλ=Πi=1maiz0,i′π,λ2+λz0,iπ,λ2. If λ∗ is a k-multiple (geometric) eigenvalue, then Rank(Cλ∗) = m−k. We may as well let the last k elements in the diagonal of Cλ∗ be zero. Hence, λ∗ is an eigenvalue of the last k problems in (33). Denote Δiλ=aiz0,i′π,λ2+λz0,iπ,λ2. We have Δiλ∗=0 and Δi′λ∗≠0 for i=m−k+1,⋯,m. Thus, λ∗ is also with algebraic multiplicity k.


## 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 A in the boundary condition.Lemma 8.
Whenm=2, the algebraic multiplicity and geometric multiplicity of λ as an eigenvalue of the problem (1) is equal.Proof.
See [6].Theorem 9.
Letm=2. If the eigenvalues of matrix A are all simple, then with finitely many exceptions, the eigenvalues of problem (1) are all simple too.Proof.
Letλn,r be an eigenvalue of problem (1). By Lemma 4,
(37)λn,r=n−1πtan−1ar+o1.
Suppose, to the contrary, that the multiplicities of the eigenvalues of problem (1) are 2. By Lemma 1, RankCλ=0. That is, Cλn,r=O2. By (13), we have
(38)cosλn,rπdiaga1,a2+sinλn,rπE2+Oeτπλn,rπ=O2.
Calculating directly and noting that the eigenvalues are real, we get(39)tanλn,rπE2=diaga1,a2+O1n.
Taking the limit asn⟶∞, (39) follows that a1=a2 which contradicts with the facts that the eigenvalues of the matrix A are all simple.
The next theorem shows the relationship between multiplicities of eigenvalues andQx when m=2 and A=cE2c≠0.Theorem 10.
Suppose thatA=cE2c≠0. If the eigenvalues of the matrix ∫0πQtdt 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 Theorem9 and therefore is omitted. Note that formulas (9) and (10) are needed to calculate Cλ in this case.



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