CUBO A Mathematical Journal Vol.10, N o ¯ 01, (93–102). March 2008 Recovering Higher-order Differential Operators on Star-type Graphs from Spectra Vjacheslav A. Yurko Department of Mathematics, Saratov University Astrakhanskaya 83, Saratov 410026, Russia yurkova@info.sgu.ru – mexmat.sgu.ru/yurko ABSTRACT We study an inverse problem of recovering arbitrary order ordinary differential operators on compact star-type graphs from a system of spectra. We establish properties of spectral characteristics, and provide a procedure for constructing the solution of the inverse problem of recovering coefficients of differential equa- tions from the given spectra. RESUMEN Estudiamos un problema inverso de recuperar el orden de operadores diferen- ciales ordinarios sobre graficos compactos de tipo estrellado a partir de un sis- tema de espectro. Propiedades de la caracteristica espectral son establecidas y es dado un procecimiento para construir la solución del problema inverso de recuperar coeficientes de ecuaciones diferenciales a partir del espectro. 94 V. Yurko CUBO 10, 1 (2008) Key words and phrases: Differential equations; Geometrical graphs; Spectral character- istics, Inverse problems, Method of spectral mappings. AMS Classification: 34A55, 34L05, 47E05. 1 Introduction We study the inverse spectral problem of recovering arbitrary order differential operators on compact star-type graphs from a system of spectra. We prove a corresponding uniqueness theorem and provide a constructive procedure for the solution of this inverse problem. For studying this inverse problem we develop the ideas of the method of spectral mappings [1]. The obtained results are natural generalizations of the well-known results on inverse problems for the differential operators on an interval ([1]-[4]). We note that boundary value problems on graphs (networks, trees) often appear in natural sciences and engineering (see [5] and the references therein). Consider a compact star-type graph T in Rm with the set of vertices V = {v0, . . . ,vp} and the set of edges E = {e1, . . . ,ep}, where v0, . . . ,vp−1 are the boundary vertices, vp is the internal vertex, and ep = [v0,vp], ej = [vp,vj ], j = 1,p − 1, e1∩. . .∩ep = {vp}. For simplicity we suppose that the length of each edge is equal to 1 (it follows from the proofs that the results remain true for arbitrary lengths of the edges). Each edge ej ∈ E is parameterized by the parameter x ∈ [0, 1]. It is convenient for us to choose the following orientation: x = 0 corresponds to the boundary vertices v0, . . . ,vp−1, and x = 1 corresponds to the internal vertex vp. An integrable function Y on T may be represented as Y (x) = {yj(x)}j=1,p, x ∈ [0, 1], where the function yj (x) is defined on the edge ej . Fix n ≥ 2. Let qν (x) = {qνj(x)}j=1,p, ν = 0,n − 2 be integrable complex-valued func- tions on T. Consider the following n-th order differential equation on T : y (n) j (x) + n−2 ∑ ν=0 qνj (x)y (ν) j (x) = λyj (x), j = 1,p, (1) where λ is the spectral parameter, qνj (x) are complex-valued integrable functions, and y (ν) j (x) ∈ AC[0, 1], j = 1,p, ν = 0,n − 1. Denote by q = {qν}ν=0,n−2 the set of the coeffi- cients of equation (1); q is called the potential. Consider the linear forms Ujν (yj ) = ν ∑ µ=0 γjνµy (µ) j (1), j = 1,p − 1, ν = 0,n − 1, where γjνµ are complex numbers, and γjν := γjνν 6= 0. The linear forms Ujν will be used in matching conditions in the internal vertex vp for for special solutions of equation (1). CUBO 10, 1 (2008) Recovering Higher-Order Differential Operators ... 95 Fix s = 1,p − 1, k = 1,n − 1, µ = k,n. Let Λskµ := {λlskµ}l≥1 be the set of the eigen- values of the boundary value problem Lskµ for equation (1) with the boundary conditions y (ν−1) s (0) = 0, ν = 1,k − 1,µ y (ξ−1) j (0) = 0, ξ = 1,n − k, j = 1,p \ s, and with the matching conditions Ujν (yj ) + y (ν) p (1) = 0, j = 1,p − 1, ν = 0,k − 1, p−1 ∑ j=1 Ujν (yj ) + y (ν) p (1) = 0, ν = k,n − 1.          (2) The inverse problem of recovering the potential from the system of spectra is formulated as follows. Inverse Problem 1. Given the spectra Λ := {Λskµ}, s = 1,p − 1, 1 ≤ k ≤ µ ≤ n, construct the potential q. This inverse problem is a generalization of the well-known inverse problems for differen- tial operators on an interval from a system of spectra (see [1-4]). For example, if n = p = 2, then Inverse Problem 1 is the classical Borg’s inverse problem of recovering Sturm-Liouville operators from two spectra. 2 Auxiliary propositions Let Ψsk(x,λ) = {ψskj (x,λ)}j=1,p, s = 1,p − 1, k = 1,n, be solutions of equation (1) satisfying the boundary conditions y (ν−1) s (0) = δkν, ν = 1,k, y (ξ−1) j (0) = 0, ξ = 1,n − k, j = 1,p \ s,    (3) and the matching conditions (2). Here and in the sequel, δkν is the Kronecker symbol. The function Ψsk is called the Weyl-type solution of order k with respect to the boundary vertex vs. We introduce the matrices Ms(λ) = [Mskν (λ)]k,ν=1,n, s = 1,p − 1, where Mskν (λ) := ψ (ν−1) sks (0,λ). It follows from the definition of ψskj that Mskν (λ) = δkν for k ≥ ν, and det Ms(λ) ≡ 1. The matrix Ms(λ) is called the Weyl-type matrix with respect to the boundary vertex vs. Denote by M = {Ms}s=1,p−1 the set of the Weyl-type matrices. 96 V. Yurko CUBO 10, 1 (2008) Let λ = ρ n . The ρ - plane can be partitioned into sectors S of angle π n ( arg ρ ∈ ( νπ n , (ν+1)π n ) , ν = 0, 2n − 1 ) in which the roots R1,R2, . . . ,Rn of the equation R n − 1 = 0 can be numbered in such a way that Re(ρR1) < Re(ρR2) < ... < Re(ρRn), ρ ∈ S. (4) We assume that the regularity condition for matching from [6] is fulfilled. The following assertion was proved in [6]. Lemma 1. Fix a sector S with the property (4). For x ∈ (0, 1), ν = 0,n − 1, s = 1,p − 1, k = 1,n, the following asymptotical formula holds ψ (ν) sks(x,λ) = ωk ρk−1 (ρRk) ν exp(ρRkx)[1], ρ ∈ S, |ρ| → ∞, where ωk := Ωk−1 Ωk , k = 1,n, Ωk := det[R ν−1 ξ ]ξ,ν=1,k, Ω0 := 1. For s = 1,p − 1, k = 1,n − 1, µ = k + 1,n, Mskµ(λ) = mkµρ µ−k [1], ρ ∈ S, |ρ| → ∞, (5) where mkµ are constants which do not depend on the potential. Let {Ckj (x,λ)}k=1,n, j = 1,p be the fundamental system of solutions of equation (1) on the edge ej under the initial conditions C (ν−1) kj (0,λ) = δkν , k,ν = 1,n. For each fixed x ∈ [0, 1], the functions C (ν−1) kj (x,λ), k,ν = 1,n, j = 1,p, are entire in λ of order 1/n. Moreover, det[C (ν−1) kj (x,λ)]k,ν=1,n ≡ 1. (6) Using the fundamental system of solutions {Ckj (x,λ)}k=1,n, one can write ψskj (x,λ) = n ∑ µ=1 Mskjµ(λ)Cµj (x,λ), j = 1,p, s = 1,p − 1, k = 1,n, (7) where the coefficients Mskjµ(λ) do not depend on x. In particular, Msksµ(λ) = Mskµ(λ), and ψsks(x,λ) = Cks(x,λ) + n ∑ µ=k+1 Mskµ(λ)Cµs(x,λ). (8) It follows from (6) and (8) that det[ψ (ν−1) sks (x,λ)]k,ν=1,n ≡ 1. CUBO 10, 1 (2008) Recovering Higher-Order Differential Operators ... 97 Fix k = 1,n, s = 1,p − 1. According to (2) and (3), Ujν (ψskj (x,λ)) + ψ (ν) skp(1,λ) = 0, j = 1,p − 1, ν = 0,k − 1, p−1 ∑ j=1 Ujν (ψskj (x,λ)) + ψ (ν) skp(1,λ) = 0, ν = k,n − 1,          (9) ψ (ν−1) sks (0,λ) = δkν, ν = 1,k, ψ (ξ−1) skj (0,λ) = 0, ξ = 1,n − k, j = 1,p \ s.    (10) Substituting the representation (7) into (9) and (10) we obtain a linear algebraic system with respect to Mskjµ(λ). Solving this system by Cramer’s rule one gets Mskjµ(λ) = ∆skjµ(λ) ∆sk(λ) , where the functions ∆skjµ(λ) and ∆sk(λ) are entire in λ of order 1/n. Thus, the functions Mskjµ(λ) are meromorphic in λ, and consequently, the Weyl-type solutions and the Weyl- type matrices are meromorphic in λ. In particular, Mskµ(λ) = ∆skµ(λ) ∆sk(λ) , s = 1,p − 1, k = 1,n − 1, µ = k + 1,n, (11) where ∆skµ(λ) := ∆sksµ(λ), ∆sk(λ) := ∆skk(λ). The function ∆skµ(λ) is the characteristic function of the boundary value problem Lskµ, and its zeros coincide with the eigenvalues Λskµ := {λlskµ}l≥1 of Lskµ. The functions ∆skµ(λ) are entire in λ of order 1/n. By Hadamard’s factorization theo- rem, the functions ∆skµ(λ) are uniquely determined up to multiplicative constants cskµ by their zeros: ∆skµ(λ) = cskµ ∞ ∏ l=1 ( 1 − λ λlskµ ) (the case when ∆skµ(0) = 0 requires evident modifications). Then, by virtue of (11), Mskµ(λ) = M 0 skµ ∞ ∏ l=1 ( 1 − λ λlskµ )( 1 − λ λlskk )−1 , s = 1,p − 1, k = 1,n − 1, µ = k + 1,n. (12) Using (5) we obtain M 0 skµ = lim |ρ|→∞ mmkρ µ−k ∞ ∏ l=1 ( 1 − λ λlskk )( 1 − λ λlskµ )−1 . (13) 98 V. Yurko CUBO 10, 1 (2008) Thus, using the given spectra Λ, one can construct uniquely the Weyl-type matrices M by (12) and (13). In other words, the following assertion holds. Theorem 1. The specification of the system of spectra Λ := {Λskµ}, s = 1,p − 1, 1 ≤ k ≤ µ ≤ n, uniquely determines the Weyl-type matrices M = {Ms}s=1,p−1 by (12)-(13). Fix s = 1,p − 1, and consider the following inverse problem on the edge es. Inverse Problem 2. Given the Weyl-type matrix Ms, construct the functions qνs, ν = 0,n − 2 on the edge es. It was proved in [6] that this inverse problem has a unique solution, i.e. the specification of the Weyl-type matrix Ms uniquely determines the potential on the edge es. Moreover, using the method of spectral mappings one can get a constructive procedure for the solution of Inverse Problem 2. It can be obtained by the same arguments as for n-th order differential operators on a finite interval (see [1, Ch.2] for details). Now we define an auxiliary Weyl-type matrix with respect to the internal vertex vp. Let ψpk(x,λ), k = 1,n, be solutions of equation (1) on the edge ep under the conditions ψ (ν−1) pk (1,λ) = δkν, ν = 1,k, ψ (ξ−1) pk (0,λ) = 0, ξ = 1,n − k. (14) We introduce the matrix Mp(λ) = [Mpkν (λ)]k,ν=1,n, where Mpkν (λ) := ψ (ν−1) pk (1,λ). Clearly, Mpkν (λ) = δkν for k ≥ ν, and det Mp(λ) ≡ 1. The matrix Mp(λ) is called the Weyl-type matrix with respect to the internal vertex vp. Consider the following inverse problem on the edge ep. Inverse Problem 3. Given the Weyl-type matrix Mp, construct the functions qνp, ν = 0,n − 2 on the edge ep. This inverse problem is the classical one, since it is the inverse problem of recovering n-th order differential equation on a finite interval from its Weyl-type matrix. This inverse problem has been solved in [1]. In particular, it is proved that the specification of the Weyl-type matrix Mp uniquely determines the potential on the edge ep. Moreover, in [1] an algorithm for the solution of Inverse Problem 3 is given, and necessary and sufficient conditions for the solvability of this inverse problem are provided. 3 Solution of the inverse problem from spectra In this section we obtain a constructive procedure for the solution of Inverse Problem 1. Our plan is the following. CUBO 10, 1 (2008) Recovering Higher-Order Differential Operators ... 99 Step 1. Using (12)-(13) construct the Weyl-type matrices M = {Ms}s=1,p−1. Step 2. Solving Inverse Problem 2 for each fixed s = 1,p − 1, we find the functions qνs, ν = 0,n − 2, s = 1,p − 1, i.e. we find the potential q on the edges e1, . . . ,ep−1. Step 3. Using the knowledge of the potential on the edges e1, . . . ,ep−1, we construct the Weyl-type matrix Mp. Step 4. Solving Inverse Problem 3 we find the functions qνp, ν = 0,n − 2, i.e. we find the potential on the edge ep. Steps 1, 2 and 4 have been already studied in Section 2. It remains to fulfil Step 3. Suppose that Steps 1-2 are already made, and we found the functions qνs, ν = 0,n − 2, s = 1,p − 1, i.e. we found the potential q on the edges e1, . . . ,ep−1. Fix s = 1,p − 1. All calcu- lations below will be made for this fixed s. Using the knowledge of the potential on the edge es, we calculate the functions Cks(x,λ), k = 1,n, and the functions ψsks(x,λ), k = 1,n, by (8). Now we are going to construct the Weyl-type matrix Mp using ψsks(x,λ), k = 1,n. Fix s = 1,p − 1. Denote zp1(x,λ) := ψs1p(x,λ) ψs1p(1,λ) . The function zp1(x,λ) is a solution of equation (1) on the edge ep, and zp1(1,λ) = 1. Moreover, by virtue of (10), one has z (ξ−1) p1 (0,λ) = 0, ξ = 1,n − k. Taking (14) into account we conclude that the solutions zp1(x,λ) and ψp1(x,λ) satisfy the same boundary conditions, and consequently, zp1(x,λ) ≡ ψp1(x,λ). Thus, ψp1(x,λ) = ψs1p(x,λ) ψs1p(1,λ) . (15) Similarly, we calculate ψpk(x,λ) = det[ψsµp(1,λ), . . . ,ψ (k−2) sµp (1,λ),ψsµp(x,λ)]µ=1,k det[ψ (ξ−1) sµp (1,λ)]ξ,µ=1,k , k = 2,n − 1. (16) Since Mpkν (λ) = ψ (ν−1) pk (1,λ), it follows from (15)-(16) that Mp1ν (λ) = ψ (ν−1) s1p (1,λ) ψs1p(1,λ) , ν = 2,n, (17) Mpkν (λ) = det[ψsµp(1,λ), . . . ,ψ (k−2) sµp (1,λ),ψ (ν−1) sµp (1,λ)]µ=1,k det[ψ (ξ−1) sµp (1,λ)]ξ,µ=1,k , (18) k = 2,n − 1, ν = k + 1,n. 100 V. Yurko CUBO 10, 1 (2008) Using the matching conditions (9) we get Ujν (ψskj ) = Usν (ψsks), 0 ≤ ν < k ≤ n − 1. (19) Since the functions ψsks were already calculated, the right-hand sides in (19) are known. For each fixed k = 1,n − 1, we successively use (19) for ν = 0, 1, . . . ,k − 1, and calculate recurrently the functions ψ (ν) skj (1,λ), k = 1,n − 1, ν = 0,k − 1, j = 1,p − 1 \ s. (20) Furthermore, it follows from (7) and (10) that Mskjµ(λ) = 0 for µ = 1,n − k, j = 1,p − 1 \ s, and consequently, ψskj (x,λ) = n ∑ µ=n−k+1 Mskjµ(λ)Cµj (x,λ), k = 1,n − 1, j = 1,p − 1 \ s. This yields ψ (ν) skj (1,λ) = n ∑ µ=n−k+1 Mskjµ(λ)C (ν) µj (1,λ), ν = 0,n − 1, k = 1,n − 1, j = 1,p − 1 \ s. (21) Fix k = 1,n − 1, j = 1,p − 1 \ s, and consider a part of the relations (21), namely, for ν = 0,k − 1. They form a linear algebraic system with respect to the functions Mskjµ(λ), µ = n − k + 1,n. Solving this system by Cramer’s rule we find these functions. Substituting them into (21) for ν ≥ k, we calculate the functions ψ (ν) skj (1,λ), k = 1,n − 1, ν = k,n − 1, j = 1,p − 1 \ s. (22) Substituting now the functions (20) and (22) into (9) we find ψ (ν) skp(1,λ), k = 1,n − 1, ν = 0,n − 1. (23) Since the functions (23) are known, one can calculate the Weyl-type matrix Mp via (17)-(18). Thus, we have obtained the solution of Inverse Problem 1 and proved its uniqueness, i.e. the following assertion holds. Theorem 2. The specification of the spectra Λ uniquely determines the potential q on T. The solution of Inverse Problem 1 can be obtained by the following algorithm. CUBO 10, 1 (2008) Recovering Higher-Order Differential Operators ... 101 Algorithm 1. Given the spectra Λ. 1) Construct the Weyl-type matrices M = {Ms}s=1,p−1 via (12)-(13). 2) Find the functions qνs, ν = 0,n − 2, s = 1,p − 1, by solving Inverse Problem 2 for each s = 1,p − 1. 3) Fix s = 1,p − 1, and calculate C (ν) ks (1,λ) for k = 1,n, ν = 0,n − 1. 4) Construct the functions ψ (ν) sks(1,λ), k = 1,n − 1, ν = 0,n − 1 by the formula ψ (ν) sks(1,λ) = C (ν) ks (1,λ) + n ∑ µ=k+1 Mskµ(λ)C (ν) µs (1,λ). 5) Find the functions ψ (ν) skj (1,λ), k = 1,n − 1, ν = 0,k − 1, j = 1,p − 1 \ s, by using the recurrent formulae (19). 6) Calculate Mskjµ(λ), k = 1,n − 1, µ = n − k + 1,n, j = 1,p − 1 \ s, by solving the linear algebraic systems n ∑ µ=n−k+1 Mskjµ(λ)C (ν) µj (1,λ) = ψ (ν) skj (1,λ), ν = 0,k − 1, for each fixed k = 1,n − 1, j = 1,p − 1 \ s. 7) Construct the functions ψ (ν) skj (1,λ), k = 1,n − 1, ν = k,n − 1, j = 1,p − 1 \ s, by the formula ψ (ν) skj (1,λ) = n ∑ µ=n−k+1 Mskjµ(λ)C (ν) µj (1,λ), ν ≥ k. 8) Find the functions ψ (ν) skp(1,λ), k = 1,n − 1, ν = 0,n − 1, by (9). 9) Calculate the Weyl-type matrix Mp via (17)-(18). 10) Construct the functions qνp, ν = 0,n − 2, by solving Inverse Problem 3. Acknowledgment. This research was supported in part by Grants 07-01-00003 and 07-01-92000-NSC-a of Russian Foundation for Basic Research and Taiwan National Science Council. Received: March 2007. Revised: September 2007. 102 V. Yurko CUBO 10, 1 (2008) References [1] V.A. Yurko, Method of Spectral Mappings in the Inverse Problem Theory, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2002. [2] V.A. Marchenko, Sturm-Liouville operators and their applications, “Naukova Dumka”, Kiev, 1977; English transl., Birkhäuser, 1986. [3] B.M. Levitan, Inverse Sturm-Liouville problems. Nauka, Moscow, 1984; English transl., VNU Sci.Press, Utrecht, 1987. [4] G. Freiling and V.A. Yurko, Inverse Sturm-Liouville Problems and their Applica- tions, NOVA Science Publishers, New York, 2001. [5] Yu.V. Pokornyi and A.V. Borovskikh, Differential equations on networks (geo- metric graphs), J. Math. Sci. (N.Y.), 119, no.6 (2004), 691–718. [6] V.A. Yurko, Recovering higher-order differential operators on star-type graphs. Schriftenreiche des Fachbereichs Mathematik, SM-DU-646, Universitaet Duisburg- Essen, 2007, 13pp. 07-cubo.pdf