# An Inverse Spectral Problem for the Matrix Sturm-Liouville Operator with a Bessel-Type Singularity

Authors: Natalia Bondarenko
Journal: International Journal of Differential Equations
 (2015)

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## Abstract

The inverse problem by the Weyl matrix is studied for the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval. We construct special fundamental systems of solutions for this equation and prove the uniqueness theorem of the inverse problem.


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## Body

## 1. Introduction

Inverse problems of the spectral analysis for systems of differential equations with Bessel-type singularities arise in quantum mechanics (e.g., in the theory of the deuteron [1]). Although inverse problems forscalar operators with singularities have been studied for a long time [2–4], there are no general results formatrix operators. In the monograph [1], the inversescattering problem is studied for some particular cases of matrix Sturm-Liouville equations with singularities. In this paper, we start a research on inversespectral problems for systems with Bessel-type singularities. We suggest an approach, based on the method of spectral mappings [5]. One of the main ingredients of our method is special fundamental systems of solutions (FSS): Bessel-type solutions with the prescribed asymptotic behavior at the singular point and Birkhoff-type solutions with the known asymptotic representations, as the spectral parameter tends to infinity. The connection between these two FSS plays the most important role in the investigation. In this paper, we formulate an inverse problem by the Weyl matrix for the matrix Sturm-Liouville operator with a singularity and prove the uniqueness theorem.Consider the matrix Sturm-Liouville equation on a finite interval with a Bessel-type singularity in the end of the interval(1)lY≔-Y′′+ωx2+QxY=λY,x∈0,T.Here Y=[yk(x)]k=1m is a vector function, λ is the spectral parameter, and Q(x) and ω are m×m matrices.We assume that the matrixω is diagonal; that is, ω=diag{ω1,ω2,…,ωm}, ωq∈R, q=1,m¯. If ω is an arbitrary Hermitian matrix, one can apply the standard unitary transform, in order to fulfill this condition. For definiteness, let ωq=νq2-1/4, ν1≥ν2≥⋯≥νm>0, νq∉N, q=1,m¯. Let the matrix function x1-2ν1Q(x) be integrable on (0,T).


## 2. Bessel-Type Solutions

Letρ=λ, argρ∈[0,π). If we put Q(x)≡0, (1) splits into m scalar equations, which have Bessel solutions [6]: (2)cjqx,λ=xμjq∑k=0∞cjkqρx2k,j=1,2,q=1,m¯,where(3)μ1q=12-νq,μ2q=12+νq,c10qc20q=12νq,cjkq=-1k·cj0q∏s=1k2s+μjq2s+μjq-1-ωq-1.Clearly, the matrix functions Cj(x,λ)=diag{cjq(x,λ)}q=1m satisfy (1).LetSj(x,λ) and Sj∗(x,λ), j=1,2, be the matrix solutions of the following integral equations:(4)Sjx,λ=Cjx,λ-∫0xGx,t,λQtSjt,λdt,Sj∗x,λ=Cjx,λ-∫0xSj∗t,λQtGx,t,λdt,where G(x,t,λ)=C2(x,λ)C1(t,λ)-C1(x,λ)C2(t,λ).Along with (1), consider the following equation:(5)-Z′′+Zωx2+Qx=λZ,x∈0,T,where Z=Z(x) is a row vector.Denote the unitm×m matrix by I, its qth column by eq, and its qth row by eq∗. The column vectors Sj(x,λ)eq, j=1,2, q=1,m¯, form a fundamental system of solutions (FSS) of (1). Similarly the row vectors eq∗Sj∗(x,λ) form an FSS of (5). Clearly, for each fixed x∈(0,T), the matrix functions Sj(ν)(x,λ),  Sj∗(ν)(x,λ) are entire in the λ-plane. Using integral equations (4) and the integrability of the matrix function x1-2ν1Q(x), one obtains the following asymptotic formulas as x→0, j=1,2, and q=1,m¯:(6)Sjx,λeq=Oxμjq,eq∗Sj∗x,λ=Oxμjq,x-μjqSjx,λ-Cjx,λeq=ox2ν1,x-μjqeq∗Sj∗x,λ-Cjx,λ=ox2ν1.Denote〈Z,Y〉:=Z′Y-YZ′. It is easy to show that if the vector functions Y(x) and Z(x) satisfy (1) and (5), respectively, then 〈Z(x),Y(x)〉 does not depend on x. Note that Cj(x,λ), j=1,2, are the matrix solutions of (5) for Q(x)≡0. Moreover, 〈Cj(x,λ),Ck(x,λ)〉=(-1)jδjkI, j,k=1,2, where δjk is the Kronecker delta. Using this fact together with asymptotic formulas (6), we obtain the relation(7)Sj∗x,λ,Skx,λ=-1jδjkI,j,k=1,2.


## 3. Birkhoff-Type Solutions

Following ideas of [3], we construct the solutions Yk(x,ρ),  k=1,2, whose columns form an FSS of (1) and have the following properties.(i1) For each fixed x∈(0,T), the matrix functions Yk(ν)(x,ρ), k=1,2, ν=0,1, are analytic in the set Ω:={ρ:Imρ>0,ρ>ρ∗} and continuous in Ω¯.(i2) For x∈(0,T), ρ∈Ω¯, ρx≥1, and ν=0,1, the following asymptotic formulas are valid:(8)Y1νx,ρ=iρνexp⁡iρxIβ,Y2νx,ρ=-iρνexp⁡-iρxIβ,where [I]β=(I+O(ρ-β)), β=min{1,2ν1}.(i3) The solutions Sj(x,λ) and Yk(x,λ) are connected by the relations(9)Ykx,λ=∑j=12Sjx,λBkjρ,and theStokes multipliersBkj(ρ) have the following asymptotic behavior:(10)Bkjρ=diag⁡ρμjqBkj0Iβ,k,j=1,2,q=1,m¯,where Bkj0=diag{βkjq} and βkjq are constant Stokes multipliers for solutions of the scalar equations -y′′+(ωq/x2)y=y (see [3]).Expansion (9) and asymptotic formula (10) play a crucial role in the study of the inverse problem. Using (9) and (10), we derive asymptotic formulas for Sj(x,λ):(11)Sjνx,λ=i-1j2-iρνexp⁡-iρxIβdiag⁡ρ-μjqB1,3-j0for ρ∈Ωδ:={ρ∈Ω:δ<argρ<π-δ}, δ>0, ν=0,1, and x∈(0,T). Similarly, Birkhoff-type solutions can be constructed for (5), and the asymptotic formulas for the solutions Sj∗(x,λ) can be obtained.


## 4. Inverse Problem

Introduce the linear formsσ1(Y):=〈S2∗(x,λ),Y〉, σ2(Y):=-〈S1∗(x,λ),Y〉. In view of (7), we have σj(Sk)=δjkI. Note that for the classical matrix Sturm-Liouville equation we have σ1(Y)=Y(0), σ2(Y)=Y′(0).Consider the boundary value problemL=L(Q,h,H) for (1) with the boundary conditions(12)UY≔σ2Y-hσ1Y=0,(13)VY≔Y′T+HYT=0,where h and H are m×m matrices. One can also take the Dirichlet-type boundary condition σ1(Y)=0 at x=0. If νm≥1/2, it is equivalent to the standard Dirichlet boundary condition Y(0)=0. Similarly, one can investigate the matrix Sturm-Liouville equation with Bessel-type singularities at both ends of the interval. Then both boundary conditions take the form similar to (12).LetΦ(x,λ) be the matrix solution of (1), satisfying the conditions U(Φ)=I, V(Φ)=0. The matrix function M(λ):=Φ(0,λ) is called theWeyl matrix of the problem L. The Weyl matrix generalizes the notion of the Weyl function for the scalar case m=1 (see [5, 7]).Letφ(x,λ) be the matrix solution of (1) under initial conditions σ1(φ)=I, σ2(φ)=h. Obviously,(14)φx,λ=S1x,λ+S2x,λh,Φx,λ=S2x,λ+φx,λMλ.Introduce the linear formsσ1∗(Z):=-〈Z,S2(x,λ)〉, σ2∗(Z):=〈Z,S1(x,λ)〉 and consider the boundary value problem L∗=L∗(Q,h,H) for (5) with the boundary conditions (15)U∗Z≔σ2∗Z-σ1∗Zh=0,V∗Z≔Z′T+ZTH=0.Similarly one can define the problem L~∗=L∗(Q~,h~,H~). Let φ∗(x,λ) and Φ∗(x,λ) be matrix solutions of (5), satisfying the conditions φ∗(0,λ)=I, φ∗′(0,λ)=h, U∗(Φ∗)=I, and V∗(Φ∗)=0. Put M∗(λ):=Φ∗(0,λ). Similarly to (14), one can easily derive(16)φ∗x,λ=S1∗x,λ+hS2∗x,λ,Φ∗x,λ=S2∗x,λ+M∗λφ∗x,λ.Note that the expressions 〈φ∗(x,λ),φ(x,λ)〉, 〈Φ∗(x,λ),Φ(x,λ)〉 do not depend on x. Using relations (14), (16), and (7), we obtain (17)φ∗x,λ,φx,λ=0,Φ∗x,λ,Φx,λ=Mλ-M∗λ.In addition,(18)Φ∗x,λ,Φx,λ=Φ∗x,λ,Φx,λx=T=V∗Φ∗ΦT,λ-Φ∗T,λVΦ=0.Hence M(λ)≡M∗(λ).We study the followinginverse problem: given the Weyl matrix M(λ), find Q(x), h, and H.Consider the boundary value problemL~=L(Q~,h~,H~) in the same form as L, but with other coefficients. We agree that if a certain symbol γ denotes an object related to L, then the corresponding symbol γ~ with tilde denotes the analogous object related to L~.Theorem 1.
IfM(λ)=M~(λ), then Q(x)=Q~(x) a.e. on (0,T), h=h~, and H=H~. Thus, the boundary value problem L can be determined by its Weyl matrix uniquely.Proof.
Define the block matrixP(x,λ)=[Pjk(x,λ)]j,k=1,2 by the formula(19)Px,λφ~x,λΦ~x,λφ~′x,λΦ~′x,λ=φx,λΦx,λφ′x,λΦ′x,λ.It follows from (14), (16), and (7) that (20)Φ∗~′x,λ-Φ~∗x,λ-φ∗~′x,λφ~∗x,λ·φ~x,λΦ~x,λφ~′x,λΦ~′x,λ=I00I.Consequently,(21)P11x,λ=φx,λΦ∗~′x,λ-Φx,λφ∗~′x,λ,P12x,λ=Φx,λφ~∗x,λ-φx,λΦ~∗x,λ.
Using (14) and (11) we obtain(22)φνx,λ=-i2-iρνexp⁡-iρxIβB120diag⁡ρ-μ1q,ρ∈Ωδ,ν=0,1,x∈0,T.In order to study the asymptotic behavior of Φ(x,λ), introduce the matrix solution ψ(x,λ) of (1) under the initial conditions at the regular end: ψ(T,λ)=-I, ψ′(T)=H. Then Φ(x,λ)=ψ(x,λ)(U(ψ))-1. Using the standard asymptotic representation for ψ(x,λ), one can expand this solution by the system Yk(x,λ), k=1,2, and then calculate asymptotic for U(ψ), applying (9) and (10). Finally, one gets the following result:(23)Φνx,λ=exp⁡iρxIβB120-1diag⁡ρ-μ2q,ρ∈Ωδ,ν=0,1,x∈0,T.Similarly one can obtain the asymptotic formulas for φ~∗(x,λ) and Φ~∗(x,λ). Substituting all these asymptotic relations into (21), we get(24)P11x,λ=Iβ,P12x,λ=Oρ-β,x∈0,T,ρ∈Ωδ.
It follows from (21), (14), and (16) that(25)P11x,λ=φx,λS2∗~′-S2x,λφ∗~′x,λ+φx,λM~∗λ-Mλφ∗~′x,λ,P12x,λ=S2x,λφ~∗x,λ-φx,λS~2∗+φx,λMλ-M~∗λφ~∗x,λ.Since M(λ)=M~(λ)=M~∗(λ), the matrix functions P11(x,λ) and P12(x,λ) are entire with respect to λ. Taking asymptotic formulas (24) into account, we conclude that P11(x,λ)≡I, P12(x,λ)≡0. By virtue of (19), φ(x,λ)≡φ~(x,λ), Φ(x,λ)=Φ~(x,λ), and therefore L=L~.



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*Source: 647396-2015-05-19.xml*