The M-matrix inverse problem for the Sturm-Liouville equation on graphs ∗ Sonja Currie † < [email protected] > Bruce A. Watson ‡ < [email protected] > School of Mathematics University of the Witwatersrand Private Bag 3, P O WITS 2050, South Africa March 7, 2009

Abstract We consider an inverse spectral problem for Sturm-Liouville boundary value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the results found in S. Currie, B.A. Watson, M-matrix asymptotics for Sturm-Liouville problems on graphs, J. Com. Appl. Math., doi: 10.1016/j.cam.2007.11.019, using the Green’s function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary value problem and are simple, located on the real axis and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions. ∗

Keywords: inverse problem, differential operators on graphs, m-function, Sturm-Liouville (2000)MSC: 34A55, 34B45, 34B20, 34L05, 34B27 . † Supported by NRF Thutuka grant no. TTK2007040500005 ‡ Supported in part by the Centre for Applicable Analysis and Number Theory. Supported by NRF grant no. FA2007041200006

1

1

Introduction

In this paper we consider the second order differential equation ly := −

d2 y + q(x)y = λy, dx2

(1.1)

where q is real-valued and continuous, on the weighted graph G with boundary conditions at the nodes formally self-adjoint with respect to l in L2 (G). For a characterisation of self-adjoint boundary value problems on graphs and formally self-adjoint boundary conditions, see [5] and [18]. Differential operators on graphs often appear in mathematics, mechanics, physics, geophysics, chemistry and engineering, see [12, 13, 20, 21, 24, 25] and the bibliographies thereof. In recent years the interest in the spectral theory of Sturm-Liouville equations on graphs has grown considerably although most of the research in this area is devoted to the so-called direct problem of studying the properties of the spectrum, see, for example [1, 29, 30]. At present there is no general theory for the spectral inverse problem for Sturm-Liouville operators on graphs. For some recent progress in the area see, [27, 28]. There have however been substantial advances on inverse spectral problems for Sturm-Liouville operators on trees, a special case of that on graphs, see [15] and [31]. In particular Brown and Weikard solve the inverse problem considered here on trees, see [4]. Various inverse spectral problems involving the recovery of the potential and boundary conditions (not dependent on the eigenparameter) for a scalar Sturm-Liouville problem were considered by Gel’fand and Levitan, Hochstadt, Krein and Mar˘cenko in [16, 19, 22, 23, 26]. We also note that recently Bennewitz, [2], gave a very elegant solution of the mfunction inverse problem. Amongst others, Binding, Browne and Watson and Chugnova, in [3, 7], consider inverse spectral problems for scalar Sturm-Liouville problems where the boundary conditions are dependent on the eigenparameter. This paper is a continuation of [11], where we considered an M-matrix associated with a system formulation of the Sturm-Liouville operator, with formally self-adjoint boundary conditions, on a graph. There, the M-matrix was related to the matrix Pr¨ ufer angle of the system boundary value problem, and, consequently, with the boundary value problem on the graph. Asymptotics for the M-matrix were obtained as the eigenparameter tended to negative infinity. It was shown that the M-matrix is a matrix Herglotz function and the boundary conditions were recovered, from the M-matrix, up to a unitary equivalence. Here we show, for co-normal boundary conditions, see appendix, that the potential can be uniquely recovered, up to a unitary equivalence, from the M-matrix, see Theorem 5.3 and Corollary 5.4. In particular instances the problem is uniquely determined, see Corollary 5.5. In section 2, prelimiaries from [11] are recalled, while, in section 3, after proving various matrix Wronskian identities and obtaining a Green’s function for the problem, we show

2

that the poles of the M-matrix are simple, located at the eigenvalues of the boundary value problem and are on the real axis. It is also shown that the residues at the poles of the M-matrix are negative semi-definite matrices of rank equal to the multiplicity of the eigenvalue, see Theorem 3.3. Norming constants and the spectral measure are considered in section 4. In particular we relate the spectral measure to the M-matrix. The inverse problem is then solved in section 5. We build on the work of Mar˘cenko, [26], (for scalar problems) to define a transformation operator which is also a Volterra operator. As a consequence, the potential can be recovered, from the M-matrix, up to a unitary equivalance for co-normal boundary conditions. The authors wish to thank the referees for their useful comments.

2

Preliminaries

We consider (1.1) on the graph G with edges ei , i = 1, . . . , K with length li respectively. From [9], equation (1.1) can be rewritten as −yi′′ (x) + qi (x)yi (x) = λyi (x),

on [0, li ],

(2.1)

where yi and qi are the restrictions of y and q, respectively, to the edge ei . Let t = lxi and y˜i (t) = equation

√1 yi (li t). li

Then, for each i = 1, . . . , K, (2.1) transforms to the

−˜ yi′′ + li2 (Qi − λ)˜ yi = 0,

t ∈ [0, 1],

(2.2)

where Qi (t) = qi (li t). The equations (2.2) for i = 1, . . . , K, are equivalent to the system LY := −W Y˜ ′′ + QY˜ = λY˜ ,

(2.3)

−2 ], Q = diag[Q1 , . . . , QK ] and Y˜ = [˜ y1 , . . . , y˜K ]T . on [0, 1], where W = diag[l1−2 , . . . , lK

The boundary conditions on the graph may be written as K X  j=1

K    X γij yj (lj ) + δij y ′ j (lj ) = 0, αij yj (0) + βij y ′ j (0) +

i = 1, ..., 2K,

(2.4)

j=1

where N is the total number of linearly independent boundary conditions, see [9]. Under the above mapping these boundary conditions transform to ˜ Y˜ ′ (0) + C˜ Y˜ (1) + D ˜ Y˜ ′ (1) = 0, A˜Y˜ (0) + B     p p 1 1 ˜ ˜ ˜ ˜ where A = [ lj αij ], B = √ βij , C = [ lj γij ] and D = √ δij . lj

lj

Let L2K denote the weighted vector L2 -space

L2K = {F : [0, 1] → CK | Fi ∈ L2 [0, 1], i = 1, . . . , K}

3

(2.5)

with the inner product < F, G >W =

K X i=1

li2

Z

1

¯ i dt = Fi G

0

Z

1

F T W −1 Gdt.

0

Here we note that < Y˜ , Z˜ >W = (y, z)G and < LY˜ , Z˜ >W = (ly, z)G . Thus the boundary value problem on the graph is formally self-adjoint in L2 (G) if and only if the system boundary value problem, (2.3), (2.5), is formally self-adjoint in L2K . The mapping, ψ : y 7→ Y˜ , with ψ : L2 (G) → L2K is an isometry, and the representation of l in L2K is ψlψ −1 = L where L is as given by (2.3), (2.5). In [9], it was shown that the formally self-adjoint boundary value problem, (2.3), (2.5), is equivalent to a formally self-adjoint boundary value problem of dimension 2K with separated boundary conditions, i.e., is equivalent to a system of the form −M Y ′′ + P Y = λY,

(2.6)

with boundary conditions A∗ Y (0) − B ∗ Y ′ (0) = 0, ∗

∗

(2.7)

′

Γ Y (1) − ∆ Y (1) = 0, (2.8) i where M = 4 diag l12 , . . . , l21 , l12 , . . . , l21 , P is a diagonal matrix dependent on the 1 1 K K     I −I 0 0 ∗ , B ∗ = √12 . Here, potential on each edge of the graph, A = √12 0 0 −I −I ˜ and ∆∗ = 2[−D ˜ B], ˜ are 2K × 2K constant matrices. Γ∗ = [C˜ A] h

The boundary value problem (2.6)-(2.8) can be rewritten as the first order system Y′ =Z

Z ′ = −G(x)Y,

and

(2.9)

with boundary conditions A∗ Y (0) − B ∗ Z(0) = 0 = Γ∗ Y (1) − ∆∗ Z(1), where G(x) = M −1 (λ − P (x)). Without loss of generality it may be assumed that the following three properties hold: (A) G(x) is continuous, real valued and symmetric. (B) A∗ B = B ∗ A and Γ∗ ∆ = ∆∗ Γ. (C) A∗ A + B ∗ B = I and Γ∗ Γ + ∆∗ ∆ = I. Here property (A) follows directly from the nature of M and P . For formally selfadjoint boundary conditions it was shown in [9, Lemma 7.1] that (B) and (C) do not pose additional constraints. In order to define the M-matrix, in [11], we needed the two solutions, W2 and W3 , of (2.6) such that   W2 (x) W3 (x) W (x) = , (2.10) W2′ (x) W3′ (x)

4

obeys the terminal condition W (1) = R, where R =



(2.11)



−Γ ∆ . ∆ Γ

The Titchmarsh-Weyl M-matrix, M = M(λ), of (2.6)-(2.8) was defined, in [11], to be the matrix M given by Ψ = W2 + W3 M, (2.12) with the constraint that Ψ obeys (2.7).

3

The nature of the poles of the M-matrix

Here we use Wronskian identities and the Green’s function to study the nature of the poles of the M-matrix. In particular in Theoerem 3.3 we show that the poles of the M-matrix are simple, located on the real axis and are the eigenvalues of (2.6)-(2.8). In addition we show that that the residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue. Lemma 3.1 For λ ∈ R, the following Wronskian identities hold: W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x) = −I,

W2∗ (x)W3′ (x) − W2′∗ (x)W3 (x) = −I, W2′∗ (x)W3 (x) − W2∗ (x)W3′ (x) = I,

W3∗ (x)W2′ (x) − W3′∗ (x)W2 (x) = I,

W2′∗ (x)W2 (x) − W2∗ (x)W2′ (x) = 0,

W3′∗ (x)W3 (x) − W3∗ (x)W3′ (x) = 0.

Proof: We prove only the first two identities, the proofs of the other four, being similar, are omitted. We begin by showing that the matrix-Wronskians W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x) and W2′∗ (x)W3 (x) − W2∗ (x)W3′ (x) are constant. Observe that [W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x)]′ = W3′′∗ (x)W2 (x) − W3∗ (x)W2′′ (x),

(3.1)

and, since W3 (x) and W2 (x) are solutions of (2.6), ¯ ∗ (x) −W3′′∗ (x)M + W3∗ (x)P = λW 3

¯ ∗ (x). − W2′′∗ (x)M + W2∗ (x)P = λW 2

and

(3.2)

Substituting (3.2) into (3.1) we obtain −1 ¯ [W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x)]′ = W3∗ (x)(P − λ)M W2 (x) − W3∗ (x)M −1 (P − λ)W2 (x).

5

−1 = M −1 (P − ¯ Now, since M −1 and P − λ are diagonal matrices and λ ∈ R, (P − λ)M λ). Therefore [W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x)]′ = 0, and W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x) is constant.

Now W3′∗ (1)W2 (1) − W3∗ (1)W2′ (1) = −Γ∗ Γ − ∆∗ ∆ = −I. Thus W3′∗ (x)W2 (x) − W3∗ (x)W2′ (x) = −I. Taking the Hermitian conjugate of the above equation gives W2∗ (x)W3′ (x) − W2′∗ (x)W3 (x) = −I. Proposition 3.2 The Green’s function for the boundary value problem (2.6)-(2.8) can be represented as  W3 (x)Ψ∗ (t)M −1 , t < x . (3.3) G(x, t) = Ψ(x)W3∗ (t)M −1 , t > x Proof: From [8, p. 24-25] the Green’s function of (2.6)-(2.8) exists. It thus remains for us to obtain (3.3). Since G(x, t) is a solution of (2.6) with respect to x for each x 6= t, we may assume  W2 (x)U1 (t) + W3 (x)U2 (t), t < x G(x, t) = W2 (x)H1 (t) + W3 (x)H2 (t), t > x, where U1 , U2 , H1 and H2 must be determined. Let Z 1 G(x, t)F (t) dt Z(x) := 0 Z Z x Z x U2 (t)F dt + W2 (x) U1 (t)F dt + W3 (x) = W2 (x) 0

0

1

H1 (t)F dt + W3 (x) x

Z

1

H2 (t)F dt. x

By definition, Z(x) is a solution of −M Z ′′ + (P − λ)Z = F,

F ∈ L2 [0, 1].

(3.4)

Differentiating Z(x) ∈ H 2 [0, 1] we obtain ′

Z (x) =

W2′ (x)

+ W3′ (x)

Z

Z

x

U1 (t)F dt + 0 1 x

W3′ (x)

Z

x

U2 (t)F dt + 0

W2′ (x)

Z

1

H1 (t)F dt x

H2 (t)F dt + W2 (x)U1 (x)F + W3 (x)U2 (x)F − W2 (x)H1 (x)F

− W3 (x)H2 (x)F which, since Z ′ (x) ∈ H 1 [0, 1] for all F ∈ L2 [0, 1], gives the condition W2 (x)U1 (x) + W3 (x)U2 (x) − W2 (x)H1 (x) − W3 (x)H2 (x) = 0.

6

(3.5)

By substituting Z ′′ (x) into (3.4), since F may vary over all of L2 [0, 1], we get the condition W2′ (x)U1 (x) + W3′ (x)U2 (x) − W2′ (x)H1 (x) − W3′ (x)H2 (x) = −M −1 .

(3.6)

Since Z must obey the boundary condition at x = 1, substituting Z(1) and Z ′ (1) into (2.8) and using (2.11) together with properties (B) and (C) of section 2 we get that Z

1

U1 (t)F dt = 0,

0

for all F ∈ L2 [0, 1]. Thus U1 ≡ 0. Similarly substituting Z(0) and Z ′ (0) into (2.7) we obtain (A∗ W2 (0) − B ∗ W2′ (0))H1 (t) = −(A∗ W3 (0) − B ∗ W3′ (0))H2 (t) a.e. on [0, 1]. Which, by [11, equation (2.15)], can be rewritten as (B ∗ W3′ (0) − A∗ W3 (0))M(λ)H1 (t) = (B ∗ W3′ (0) − A∗ W3 (0))H2 (t), a.e. on [0, 1]. Since B ∗ W3′ (0) − A∗ W3 (0) is invertible away from the eigenvalues MH1 (t) = H2 (t), a.e. on [0, 1] for λ not an eigenvalue. Thus (3.5) and (3.6) become W3 (x)U2 (x) − W2 (x)H1 (x) − W3 (x)MH1 (x) = 0

(3.7)

W3′ (x)U2 (x) − W2′ (x)H1 (x) − W3′ (x)MH1 (x) = −M −1 ,

(3.8)

and respectively. Multiplying (3.7) by W2′∗ (x) and (3.8) by W2∗ (x) and subtracting the resulting equations gives U2 − (W2′∗ W2 − W2∗ W2′ )H1 − (W2′∗ W3 − W2∗ W3′ )MH1 = W2∗ M −1 a.e. on [0, 1] which by Lemma 3.1 gives U2 (x) = MH1 (x) + W2∗ (x)M −1 . Similarly multiplying (3.7) by W3′∗ (x) and (3.8) by W3∗ (x), subtracting the resulting equations and using Lemma (3.1) gives H1 (x) = W3∗ (x)M −1 .

7

Thus G(x, t) =



W3 (x)(MW3∗ (t) + W2∗ (t))M −1 , t < x . (W2 (x) + W3 (x)M)W3∗ (t)M −1 , t > x

For λ ∈ R, M(λ) = M∗ (λ) since M (λ) is a Herglotz function, see [11]. So by (2.12) we get  W3 (x)Ψ∗ (t)M −1 , t < x G(x, t) = . Ψ(x)W3∗ (t)M −1 , t > x In [11] it was shown that the M-matrix defined in (2.12) is a Herglotz function and as such it admits the following representation, see [17],

M(λ) = C + Dλ +

X

Mn

λn



1 λn − λn − λ 1 + λ2n



,

1 where C = Re(M(i)) and D = limη→∞ ( iη M(iη)) ≥ 0. Thus

lim (λ − λn )M(λ) =

λ→λn

lim (λ − λn )(C + Dλ) + lim (λ − λn ) λ→λn

λ→λn

= −Mn .

X

Mn

λn



1 λn − λn − λ 1 + λ2n

We now give the main theorem of this section. Theorem 3.3 The poles of the M-matrix are simple, located on the real axis and are the eigenvalues of (2.6)-(2.8). The residue at a pole is a negative semi-definite matrix of rank equal to the multiplicity of the eigenvalue. Proof: By [17], since M(λ) is a matrix-valued Herglotz function, it follows that the poles of the M-matrix are simple and located on the real axis and that the residue at a pole is a negative semi-definite matrix. Also, from [11, Proposition 2.2], all the poles of M are eigenvalues of (2.6)-(2.8). At an eigenvalue of (2.6)-(2.8) the Green’s function of (2.6)(2.8) has a pole, see representation (3.12), giving that if λ is an eigenvalue of (2.6)-(2.8) then λ is a pole of G(x, t) and thus, by (3.3), λ is a pole of Ψ and hence, by (2.12), is a pole of M. Therefore it remains to show that −Mn , the residue of the pole of M(λ) at λn , has rank equal to the multiplicity of the eigenvalue. From (2.12) (λ − λn )Ψ(x, λ) = (λ − λn )W2 (x, λ) + (λ − λn )W3 (x, λ)M(λ) and taking the limit as λ tends to λn gives lim (λ − λn )Ψ(x, λ) = W3 (x, λn ) lim (λ − λn )M(λ). λ→λn

λ→λn

8



Thus lim (λ − λn )Ψ(x, λ) = −W3 (x, λn )Mn .

(3.9)

λ→λn

By taking the Hermitian transpose of (3.9) we obtain lim (λ − λn )Ψ∗ (x, λ) = −Mn∗ W3∗ (x, λn ).

λ→λn

(3.10)

By proposition 3.2 we have that the Green’s function for the boundary value problem (2.6)-(2.8) can be represented as  W3 (x, λ)Ψ∗ (t, λ)M −1 , t < x G(x, t) = Ψ(x, λ)W3∗ (t, λ)M −1 , t > x. Thus using (3.9) and (3.10) 

(λ − λn )W3 (x, λ)Ψ∗ (t, λ)M −1 , t < x lim (λ − λn )Ψ(x, λ)W3∗ (t, λ)M −1 , t > x λ→λn  W3 (λn , x)(limλ→λn (λ − λn )Ψ∗ (t, λ))M −1 , t < x = (limλ→λn (λ − λn )Ψ(x, λ))W3∗ (t, λn )M −1 , t > x  W3 (λn , x)Mn∗ W3∗ (t, λn )M −1 , t < x = − W3 (λn , x)Mn W3∗ (t, λn )M −1 , t > x.

lim (λ − λn )G(x, t) =

λ→λn

Hence lim (λ − λn )G(x, t) = −W3 (λn , x)Mn W3∗ (t, λn )M −1

λ→λn

(3.11)

for all t since, as M(λ) is a Herglotz function, M∗ (λ) = M(λ), for λ ∈ R, giving Mn∗ = Mn . We now observe that, by [8], the Green’s function of the boundary value problem (2.6)(2.8) is also given by Z 1 νn ∞ X X < F, Fn,j > G(x, t)F (t) dt = Fn,j (x), (3.12) λn − λ 0 n=1 j=1

for F ∈ L2 [0, 1], where νn is the multiplicity of the eigenvalue λn , Fn,j , j = 1, . . . , νn , is an orthonormal sequence of eigenfunctions corresponding to λn (λn not repeated according to multiplicity). Recall that the inner product is given by Z 1 Z 1 2K X 2 ¯ li F T M −1 Gdt, Fi Gi dt = < F, G >= (3.13) i=1

0

0

where li = lK+i for i = 1, . . . , K.

Since F, Fn,j are column vectors and M is a diagonal matrix F T (t)M −1 F¯n,j (t)Fn,j (x) = ∗ (t)M −1 F (t)F (x). But F ∗ (t)M −1 F (t) is scalar so F ∗ (t)M −1 F (t) and F (x) Fn,j n,j n,j n,j n,j commute to give Z 1X Z 1 ∞ X νn 1 ∗ Fn,j (x)Fn,j (t)M −1 F (t) dt, G(x, t)F (t) dt = λn − λ 0 0 n=1 j=1

9

for all F ∈ L2 [0, 1]. Hence, G(x, t) =

∞ X νn X n=1 j=1

1 ∗ Fn,j (x)Fn,j (t)M −1 . λn − λ

Taking the limit as λ → λn we get νn X

lim (λ − λn )G(x, t) = −

λ→λn

∗ Fn,j (x)Fn,j (t)M −1 .

j=1

Thus by (3.11) νn X

∗ Fn,j (x)Fn,j (t)M −1 = W3 (λn , x)Mn W3∗ (t, λn )M −1 .

(3.14)

j=1

Since Fn,j is an eigenfunction corresponding to λn , it can be written as Fn,j (x) = W3 (λn , x)cn,j

(3.15)

where cn,j is a column vector. Substituting (3.15) into (3.14) gives   νn X cn,j c∗n,j − Mn  W3∗ (λn , t) = 0. W3 (λn , x) 

(3.16)

j=1

Differentiating (3.16) with respect to x we obtain   νn X cn,j c∗n,j − Mn  W3∗ (λn , t) = 0. W3′ (λn , x) 

(3.17)

j=1

Pre-multiplying (3.16) by W2′∗ (λn , x) and (3.17) by W2∗ (λn , x) and subtracting the resulting equations gives   νn X cn,j c∗n,j − Mn  W3∗ (λn , t) W2′∗ (λn , x)W3 (λn , x)  j=1



−W2∗ (λn , x)W3′ (λn , x) 

νn X j=1



cn,j c∗n,j − Mn  W3∗ (λn , t) = 0.

Now applying Lemma 3.1 we obtain   νn X  cn,j c∗n,j − Mn  W3∗ (λn , t) = 0. j=1

10

(3.18)

Differentiating (3.18) with respect to t gives   νn X  cn,j c∗n,j − Mn  W3′∗ (λn , t) = 0.

(3.19)

j=1

Pre-multiplying (3.18) by W2′∗ (λn , x) and (3.19) by W2∗ (λn , t), subtracting the resulting equations and using lemma 3.1 gives νn X

cn,j c∗n,j = Mn ,

j=1

and as already noted −Mn is a negative semi-definite matrix. Now define the 2K × 2K matrix Cn := [cn,1 , . . . , cn,νn , 0, . . . , 0]. Then since cn,1 , . . . , cn,νn are linearly independent the rank of Cn is νn . Also the rank of Cn is equal to the number of non-zero eigenvalues of Cn , counted by multiplicity. Denote these eigenvalues by µ1 , . . . , µνn . Then Cn Cn∗ has non-zero eigenvalues |µ1 |2 , . . . , |µνn |2 . Thus Cn Cn∗ has rank νn and Rank(−Mn ) = νn .

4

Norming constants and spectral measure

In this section we obtain expressions for the norming constants associated with the boundary value problem (2.6)-(2.8). We then give a form for the spectral measure in terms of the norming constants and show how it relates to the M-matrix. We define the norming constant Hn by Z 1 W3∗ (λn , x)M −1 W3 (λn , x) dx. Hn = 0

It should be noted that the norming constants given here are consistent with those found using methods analogous to those of Freiling and Yurko, in [14]. From (3.15) and the definition of Fn,j we have that < W3 (λn , x)cn,j , W3 (λn , x)cn,i >= δi,j , which, by (3.13), implies that Z

1

cTn,j W3T (λn , x)M −1 W3 (λn , x)cn,i dx = δi,j .

0

11

Hence c∗n,j Hn cn,i

=

c∗n,j

Z

1

W3∗ (λn , x)M −1 W3 (λn , x) dx

0

So if Cn = [cn,1 , . . . , cn,νn ] then



cn,i = δi,j .

(4.1)

Cn∗ Hn Cn = Iνn .

Also note that < W3 (λn , x)cn,j , W3 (λm , x)cm,k >= 0 for all n 6= m. Theorem 4.1 Let the resolution function, ρ, be given by lim ρ(µ) = 0 and µ→−∞ Z µ2 1 ρ(µ2 ) − ρ(µ1 ) = lim (M(u + iδ) − M∗ (u + iδ)) du, 2πi δ→0+ µ1

(4.2)

where µ1 < µ2 are not eigenvalues of (2.6)-(2.8). Then X ρ(µ) = − Res λ=λi (M(λ)) λi <µ

and ρ is the spectral measure associated with (2.6)-(2.8), i.e. X ρ(µ) = Mi λi <µ

with f (x) = for all f ∈ L2 [0, 1].

Z

∞

−∞



W3 (λ, x)dρ(λ)

Z

1

W3∗ (λ, τ )M −1 f (τ ) dτ, 0



,

(4.3)

Proof: For µ ∈ R, since M(µ) is a Herglotz function, M∗ (µ) = M(µ). Thus if there is no eigenvalue in [µ1 , µ2 ] we obtain Z µ2 Z µ2 ∗ (M(u) − M∗ (u)) du = 0, lim (M(u + iδ) − M (u + iδ)) du = δ→0+

µ1

µ1

as expected for a spectral measure. Since the spectrum is bounded below, this also gives that ρ(µ) = 0 for all µ < λ0 . Note that since M(z) is a Herglotz function M(z) = M∗ (¯ z ), for z ∈ C, giving that M(u + iδ) = M∗ (u − iδ). Hence for a single eigenvalue of (2.6)-(2.8), λ0 ∈ (µ1 , µ2 ), we have Z µ2 Z µ2 ∗ lim (M(u + iδ) − M (u + iδ)) du = lim (M(u + iδ) − M(u − iδ)) du δ→0+ µ1 δ→0+ µ1 Z M(λ0 + z) dz = C1 Z M(λ0 + z) dz = − C2

= −2πiRes

where C1 and C2 are as given in the diagram below.

12

λ=λ0 (M(λ))

 ?

µ1 r −iδ

? ?





6iδ 6

 r 

r µ2

0 

λ

≡

µ1 ?

6

-

 r

 C1 

λ0

-

µ2

6

C2

Therefore, for general µ not an eigenvalue we get X X ρ(µ) = −Res λ=λi (M(λ)) = Mi . λi <µ

λi <µ

Now by (4.1),  Z 1 Z 1 ∗ −1 ∗ (W3 (λn , x)cn,j )∗ M −1 (W3 (λm , x)cm,i ) dx cn,j W3 (λn , x)M W3 (λm , x) dx cm,i = 0

0

= δi,j δm,n .

(4.4)

Since (W3 (λn , τ )cn,j )∗ M −1 f (τ ) is a scalar, the spectral projection Eλn [f ] of f is given by νn Z 1 X (W3 (λn , τ )cn,j )∗ M −1 f (τ ) dτ W3 (λn , x)cn,j Eλn [f ](x) = =

j=1 0 νn X

W3 (λn , x)cn,j c∗n,j

j=1

for all f ∈ L2 [0, 1]. Thus Eλn [f ](x) = W3 (λn , x)

νn X

cn,j c∗n,j

j=1

Z

Z

1

W3∗ (λn , τ )M −1 f (τ ) dτ 0

1

W3∗ (λn , τ )M −1 f (τ ) dτ 0

giving Eλn [f ](x) = W3 (λn , x)Mn

Z

1

Z

1

W3∗ (λn , τ )M −1 f (τ ) dτ.

(4.5)

0

Hence f (x) =

∞ X

W3 (λn , x)Mn

n=0

W3∗ (λn , τ )M −1 f (τ ) dτ 0

and thus (4.3) holds for all f ∈ L2 [0, 1].

5

Recovery of the operator

In [26], Mar˘cenko considered an inverse spectral problem for scalar Sturm-Liouville boundary value problems. In this, the main section of the paper, we build on the

13

method of Mar˘cenko to recover the potential for Sturm-Liouville boundary value problems on a graph from the M-matrix. Boundary conditions are recovered using M-matrix asymptotics, see [11]. From Mar˘cenko, [26, p. 30, Theorem 1.2.2] we have as a direct consequence the following lemma: Lemma 5.1 There exists a kernel, kh,m,q (t, y), (k∞,m,q (t, y) resp.) such that, vh,m,q [f ](t) :=

Z

1

kh,m,q (t, y)f (y) dy, t

(v∞,m,q [f ](t) :=

Z

1 t

k∞,m,q (t, y)f (y) dy resp.)

defines a continuous linear transformation on L2 [0, 1], and if yλ is the solution of −myλ′′ = λyλ on [0, 1] with yλ′ (1) = hyλ (1) (yλ (1) = 0 resp.), for m > 0 a real constant, then zλ := (I + vh,m,q )yλ (zλ := (I + v∞,m,q )yλ resp.) is the solution of −mzλ′′ + qzλ = λzλ on [0, 1] with zλ′ (1) = hyλ (1) and zλ (1) = yλ (1) (zλ (1) = 0 and zλ′ (1) = yλ′ (1) resp.), for each λ ∈ R. ˜ ∗, ∆ ˜ ∗ , P˜ ) the boundLet (Γ∗ , ∆∗ , P ) denote the boundary value problem (2.6)-(2.8) and (Γ ˜ ∆ by ∆ ˜ and P by P˜ . ary value problem (2.6)-(2.8) but with Γ replaced by Γ, ˜∗, ∆ ˜ ∗ , P˜ ) have the same M-matrix. If Lemma 5.2 Let the problems (Γ∗ , ∆∗ , P ) and (Γ there exists a linear continuous transformation operator, H, on L2 [0, 1], independent of λ, which maps ˜3 (λ, x), H[W3 (λ, x)] = W (5.1) ˜3 (λ, x) is the solution to (Γ ˜∗, ∆ ˜ ∗ , P˜ ) obeying W ˜3 (λ, 1) = ∆ ˜ and W ˜3 ′ (λ, 1) = Γ, ˜ where W then H is unitary. Proof: Consider f ∈ L2 [0, 1], then as in (4.3), H[f ](x) = = = =

∞ X

n=1 ∞ X

n=1 ∞ X

n=1 ∞ X

n=1

 Z H W3 (λn , x)Mn

1

Z

1

W3∗ (λn , τ )M −1 f (τ ) dτ

0

H[W3 (λn , x)]Mn



W3∗ (λn , τ )M −1 f (τ ) dτ

0

H[W3 (λn , x)]

νn X

cn,j c∗n,j

˜3 (λn , x) W

cn,j c∗n,j

Z

0

j=1

14

1

W3∗ (λn , τ )M −1 f (τ ) dτ

0

j=1

νn X

Z

1

W3∗ (λn , τ )M −1 f (τ ) dτ.

˜ and H[f ] ∈ L2 [0, 1]. Since M(λ) = M(λ) for all λ ∈ R, we have νn X

cn,j c∗n,j

˜n = = Mn = M

0

=

(H[f ])∗ M −1 H[f ] dx ∗ X Z 1 X ∞ Z 1 νn −1 ˜3 ∗ (λn , x) cn,j c∗n,j W W3 (λn , τ )M f (τ ) dτ 0 m,n=1

0

j=1

˜3 (λm , x) × M −1 W =

c˜n,j c˜∗n,j .

j=1

j=1

Hence R1

νn X

∞ X

m,n=1 νm X

×

Z

1

νm X

cm,i c∗m,i

0

1

W3 (λm , s)M −1 f (s) ds dx

0

i=1

W3 (λn , τ )M

Z

−1

f (τ ) dτ

∗ X νn

c˜n,j c˜∗n,j

j=1

c˜m,i c˜∗m,i

i=1

Z

1

Z

1

˜3 ∗ (λn , x)M −1 W ˜3 (λn , x) dx W

0

W3 (λm , s)M −1 f (s) ds. 0

˜3 (λn , x), so Now (4.4) also holds for cn,j replaced by c˜n,j and W3 (λn , x) by W c˜∗n,j

Z

1

˜3 (λn , x) dx c˜m,i = δi,j δm,n ˜3 ∗ (λn , x)M −1 W W

0

˜ n , gives which, along with Mn = M R1 (H[f ])∗ M −1 H[f ] dx 0 ∗ Z ∞ X νn Z 1 X = W3 (λn , τ )M −1 f (τ ) dτ cn,j c∗n,j =

n=1 j=1 2

1

W3 (λn , s)M −1 f (s) ds

0

0

||f || .

Therefore ||H[f ]||2 = ||f ||2 for arbitrary f ∈ L2 [0, 1]. Hence H is unitary in L2 [0, 1]. We will now use Lemma 5.1 and Lemma 5.2 to prove the main result of the paper. ˜∗, ∆ ˜ ∗ , P˜ ) have the same M-matrix, i.e. Theorem 5.3 If the problems (Γ∗ , ∆∗ , P ) and (Γ ˜ ˜ and Γ = U Γ. ˜ Here U = ΓΓ ˜ ∗ + ∆∆ ˜ ∗ is M(λ) = M(λ) for all λ ∈ R, then ∆ = U ∆ a unitary matrix. In addition, if we assume that the boundary conditions, (2.8), are co-normal, i.e. Γ and ∆ can be written in the form given in Theorem 6.1, and that the ˜ ∆, ˜ then P = U P˜ U ∗ . weight matrix M commutes with Γ, ∆, Γ,

15

˜ and Γ = U Γ, ˜ where U is as defined in the statement Proof: From [11] we have ∆ = U ∆ of the theorem. ˜ ∆ ˜ and that the In order to prove P = U P˜ U ∗ we assume that M commutes with Γ, ∆, Γ, boundary conditions, (2.8), are co-normal, i.e. Γ and ∆ can be written in the form given in Theorem 6.1. Then the solution Y¯ (t) to −M Y¯ ′′ = λY¯ obeying (2.8) is given by √ √ 1 1 1 1 Y¯ (t) = cos( λM − 2 (t − 1))∆δ + √ sin( λM − 2 (t − 1))M 2 Γδ λ p   α1 1 + |µ1 |2     p ...  αn 1 + |µn |2   and γi , i = n + 1, . . . , 2K are as in Theorem 6.1. where δ =    γn+1     ... γ2K

Now, since M commutes with Γ and ∆, M commutes with w∗k , see Appendix, giving   √ √ 1 1 ∗ − 21 − 12 ¯ Y · wk = wk cos( λM (t − 1))∆δ + √ sin( λM (t − 1))M 2 Γδ λ √ √ 1 1 1 1 = w∗k ∆ cos( λM − 2 (t − 1))δ + w∗k Γ √ sin( λM − 2 (t − 1))M 2 δ. λ Thus for k = 1, . . . , n Y¯ · wk = p

1 1 + |µk |2



 1 √ −1 √ −1 1 2 2 2 cos( λmk (t − 1)) + mk µk √ sin( λmk (t − 1)) δk λ

and for k = n + 1, . . . , 2K 1 1 √ −1 Y¯ · wk = mk2 √ sin( λmk 2 (t − 1))δk λ

where since M is diagonal mk is the kth diagonal entry of M and δk is the kth entry of δ. So for k = 1, . . . , n we have that Y¯ · w k (1) = αk and (Y¯ · w k )′ (1) = µk αk and for k = n + 1, . . . , 2K we have Y¯ · w k (1) = 0 and (Y¯ · w k )′ (1) = γk meaning that we can now use lemma 5.1 with h = µk . For k = n + 1, . . . , 2K, by lemma 5.1 there exists a kernel, k∞,mk ,qk (t, y) := K k (t, y) such that, Z 1 k K k (t, y)f (y) dy, V [f ](t) := t

defines a Volterra map which is also a continuous linear transformation on L2 [0, 1], and since Y¯ · w k is the solution of −mk (Y¯ · w k )′′ = λ(Y¯ · w k ) on [0, 1] with Y¯ · wk (1) = 0, then Zk := (I + V k )(Y¯ · w k )

16

is the solution of −mk Zk′′ +qk Zk = λZk on [0, 1] with Zk (1) = 0 and Zk′ (1) = (Y¯ ·w k )′ (1) = γk , for each λ ∈ R. Also for k = 1, . . . , n there exists a kernel, kµk ,mk ,qk (t, y) := K k (t, y) such that, k

V [f ](t) :=

Z

1

K k (t, y)f (y) dy, t

defines a Volterra map which is also a continuous linear transformation on L2 [0, 1], and since Y¯ · wk is the solution of −mk (Y¯ · w k )′′ = λ(Y¯ · wk ) on [0, 1] with Y¯ · w ′k (1) = µk Y¯ · wk (1), then Zk := (I + V k )(Y¯ · w k )

is the solution of −mk Zk′′ + qk Zk = λZk on [0, 1] with Zk′ (1) = µk Y¯ · wk (1) = αk µk and Zk (1) = Y¯ · wk (1) = αk , for each λ ∈ R. Thus

¯ Z(t) :=

2K X

wk Zk (t)

k=1

=

2K X k=1

=

wk Y¯ (t) · w k +

Y¯ (t) +

Z

t

2K 1X

2K X k=1

wk

Z

1 t

K k (t, y)Y¯ (y) · w k dy

K k (t, y)w k w ∗k Y¯ (y) dy

k=1

:= (I + VP,M )Y¯ (t), where VP,M is a Volterra map. Hence if Y¯ (t) is the solution of −M Y¯ (t)′′ = λY¯ (t) with Y¯ (1) = ∆ and Y¯ ′ (1) = Γ, then ¯ = Y¯ (t) + Z(t)

Z

t

2K 1X

K k (t, y)w k w∗k Y¯ (y) dy

k=1

¯ ′′ + P Z(t) ¯ = λZ(t) ¯ ¯ is the solution of −M Z(t) with Z(1) = ∆ and Z¯ ′ (1) = Γ. We note that (I + VP,M )−1 = I + WP,M , where WP,M is Volterra, see [26, p. 26]. Thus, if Y is the solution of −M Y ′′ + P Y = λY on [0, 1] with Y (1) = ∆ and Y ′ (1) = Γ then Z := (I + WP,M )Y is the solution of −M Z ′′ = λZ on [0, 1] with Z(1) = ∆ and Z ′ (1) = Γ. Hence Z˜ := U ∗ (I + WP,M )Y = U ∗ Z

17

˜ ˜ and is the solution of −U ∗ M U (U ∗ Z)′′ = λ(U ∗ Z) with Z(1) = U ∗ Z(1) = U ∗ ∆ = ∆ ′ ∗ ′ ∗ ˜ ˜ ˜ Z (1) = U Z (1) = U Γ = Γ. Since U M = M U we get that Z is the solution of ˜ ˜ and Z˜ ′ (1) = Γ. ˜ −M Z˜ ′′ = λZ˜ with Z(1) =∆ Let

Y˜ := (I + VP˜ ,M )Z˜ = (I + VP˜ ,M )U ∗ (I + WP,M )Y,

˜ and Y˜ ′ (1) = Γ. ˜ then Y˜ is the solution of −M Y˜ ′′ + P˜ Y˜ = λY˜ with Y˜ (1) = ∆ Let HY := (I + VP˜ ,M )U ∗ (I + WP,M )Y for Y ∈ L2 [0, 1]. If Y is any solution of −M Y ′′ + P Y = λY , then Y˜ := HY is the solution of −M Y˜ ′′ + P˜ Y˜ = λY˜ with Y˜ (1) = U ∗ Y (1) ˜ 3 and HW2 = W ˜ 2 ,, and hence, by Lemma and Y˜ ′ (1) = U ∗ Y ′ (1). In particular HW3 = W 5.2, H is unitary. Let, N := U H, then N := U (I + VP˜ ,M )U ∗ (I + WP,M ) = I + WP,M + U VP˜ ,M U ∗ + U VP˜ ,M U ∗ WP,M = I + J, where J is Volterra. We now show N is unitary. For g, f ∈ L2 [0, 1], we have < g, N ∗ f >=< N g, f >, which implies Z

But N = U H so Z

1 T

g M

−1

0

N ∗f

dt =

Z

g M

−1

(U H)∗ f

(N g)T M −1 f¯ dt.

0

1 T

1

dt =

0

= =

Z

Z

Z

1

(U (Hg))T M −1 f¯ dt

0 1

(Hg)T U T M −1 f¯ dt

0 1 0

(Hg)T U ∗ M −1 f dt.

Since M U = U M we have M −1 U ∗ = U ∗ M −1 . Hence, Z 1 Z Z 1 (Hg)T M −1 U ∗ f dt = gT M −1 (U H)∗ f dt = 0

0

1

gT M −1 H ∗ (U ∗ f ) dt. 0

Thus (U H)∗ = H ∗ U ∗ = H −1 U ∗ , where the latter equality is due to H being unitary. Hence (U H)(U H)∗ = I and (U H)∗ (U H) = I. Therefore N is unitary. But N = I + J where J is Volterra and N is unitary, so by [6, p.93], we have J = 0 and ˜ 3 = U ∗ W3 and W ˜ 2 = U ∗ W2 , N = I, giving U ∗ = (I + VP˜ ,M )U ∗ (I + WP,M ). Therefore W and −M (U ∗ Wj )′′ + P˜ U ∗ Wj = λU ∗ Wj , (5.2) for j = 2, 3.

18

Premultiplying equation (5.2) by U and noting that M U = U M , gives −M Wj′′ + U P˜ U ∗ Wj = λWj , for j = 2, 3, but we also have that −M Wj′′ + P Wj = λWj , for j = 2, 3 . So

(U P˜ U ∗ − P )Wj = 0,

for j = 2, 3, and for all d ∈ R4K , (U P˜ (t)U ∗ − P (t))[W2 (t), W3 (t)]d = 0 for all t ∈ [0, 1]. But the column space of [W2 , W3 ] spans the solution space of −M Y ′′ + P Y = λY , so for each t0 ∈ [0, 1] and c ∈ R2K there exists d ∈ R4K such that c = [W2 (t0 ), W3 (t0 )]d giving (U P˜ (t0 )U ∗ − P (t0 ))c = 0 for each c ∈ R2K . Thus P = U P˜ U ∗ . ˜ n and not that the entire Remark In fact, for the above result, we only require Mn = M M-matrices are equal. The following two corollaries are immediate consequences of the above theorem. Corollary 5.4 If all the edges of the graph G have the same length, l, and the systems ˜∗, ∆ ˜ ∗ , P˜ ) have the same M-matrix, then ∆ = U ∆, ˜ Γ = UΓ ˜ problems (Γ∗ , ∆∗ , P ) and (Γ ∗ ∗ ∗ ˜ ˜ ˜ and P = U P U where U = ΓΓ + ∆∆ is a unitary matrix. Proof: Since all the edges of the graph G have the same length l we have that M = ˜ ∆. ˜ The result now follows from Theorem 5.3. which commutes with Γ, ∆, Γ,

4 I l2

Corollary 5.5 If the problems (Γ∗ , ∆∗ , P ) and (Γ∗ , ∆∗ , P˜ ) have the same M-matrix and M commutes with Γ, ∆ then P = P˜ . ˜ = Γ, ∆ ˜ = ∆ and hence U = I. The result Proof: In the notation of Theorem 5.3, Γ follows immediately from Theorem 5.3. ′ (1) = Note that given a set of eigenvalues λn and the terminal value, ∆∗ Fn,j (1) + Γ∗ Fn,j ∗ ∗ ′ ∆ W3 (λn , 1)cn,j + Γ W3 (λn , 1)cn,j = cn,j , Mn is uniquely determined and the above corollaries apply.

Remark The above note is actually a more appealing result since it means that from the eigenvalues and the data at the nodes of the given graph, i.e. the terminal conditions, we can recover the boundary conditions and the potential. It does not rely on the superficial nodes inserted into each edge only on the original given nodes.

19

6

Appendix

The definition of co-normal boundary conditions on a graph is given in [10]. An immediate consequence of this is that for the system formulation (2.6)-(2.8) the boundary conditions (2.8), at x = 1, are co-normal if and only if Γ and ∆ are such that, when    C2K   u ∗ ∗ ′ ⊕ | Γ u S= ∈ = 0 , − ∆ u  u′  2K C   u 2K is such that there exists a subspace N , of dimension n, of C so that ∈ S for 0 all u  ∈ N and there exists a real diagonal matrix D =: diag{d1 , . . . , d2K } such that  u ∈ S if and only if u ∈ N and (Du − u′ ) · v = 0 for all v ∈ N . u′ We remark that co-normal boundary conditions on a graph correspond in nature to conormal (non-oblique) boundary conditions for elliptic partial differential operators. Most physically interesting boundary conditions on graphs fall into the co-normal category. In particular, ‘Kirchhoff’, Dirichlet, Neumann and periodic boundary conditions are all co-normal, but this class does not include all self-adjoint boundary-value problems on graphs. For example consider a single loop, i.e. the interval [0, 1] where the boundary conditions at 0 and at 1 are connected as follows y(0) = y ′ (1) and y(1) = −y ′ (0). These boundary conditions give a self-adjoint boundary-value problem with non co-normal boundary conditions. Theorem 6.1 Suppose that the boundary conditions, (2.8), are co-normal and that S, N , D are as given above. Then there exists an orthonormal basis w1 , . . . , w 2K for C2K and real numbers µ1 , . . . , µn such that, without loss of generality, ∆ and Γ may be written as "

and

∆= p "

Γ= p

w1 1 + |µ1 |2

µ1 w 1

1 + |µ1 |2

,..., p

,..., p

wn 1 + |µn |2

µn w n

1 + |µn |2

#

, 0, . . . , 0

(6.1) #

, w n+1 , . . . , w 2K .

(6.2)

Proof: Let v 1 , . . . , v n be an orthonormal basis for N and let v n+1 , . . . , v n+m , where 2K n  + m= 2K, be the extension of this basis for N to an orthonormal basis for C . Now u ∈ S if and only if u′ u=

n X i=1

αi v i

and

Du − u′ = −

20

2K−n=m X j=1

βj v j+n

for α1 , . . . , αn ∈ C and β1 , . . . , βm ∈ C. I.e. u′ = Du +

m X

βj v j+n =

n X

αi Dv i +

i=1

j=1

m X

βj v j+n .

j=1

In particular we need ′

u · vk =

n X i=1

αi (Dv i ) · v k +

So for k = 1, . . . , n, we get ′

u · vk = and for k = n + 1, . . . , 2K we get u′ · v k =

m X

n X i=1

n X i=1

βj v j+n · v k ,

j=1

k = 1, . . . , 2K.

αi (Dv i ) · v k ,

αi (Dv i ) · v k + βk .

Since βk are arbitrary we can set n X i=1

αi (Dv i ) · v k + βk := γk

giving γk an arbitrary element of C and  Pn i=1 αi (Dv i ) · v k , k = 1, . . . , n u′ · v k = γk , k = n + 1, . . . , 2K. Thus u′ = =

n X

i,k=1 n X

i,k=1

v k ((Dv i ) · v k )αi + v k dk,i αi +

m X

m X

γk+n v k+n

k=1

γk+n v k+n

k=1

where dk,i = (Dv i ) · v k . Since D is real and diagonal, [dk,i ] is Hermitian giving that [dk,i ] is diagonalizable and the eigenvalues are semisimple. Hence [dk,i ] has real eigenvalues µ1 , . . . , µn , say, with orthonormal eigenvectors say z 1 , . . . , z n ∈ Cn . I.e. [dk,i ]z j = µj z j = z j µj giving [dk,i ][z 1 , . . . , z n ] = [z 1 , . . . , z n ][µi ] where [µi ] = diag[µ1 , . . . , µn ].

21

Therefore n X

n X

v k dk,i αi =

i,k=1

v k dk,i (u · v i )

i,k=1



 v ∗1 u = [v 1 , . . . , v n ][dk,i ]  . . .  v ∗n u



 v ∗1 u = [v 1 , . . . , v n ][z 1 , . . . , z n ][µi ][z 1 , . . . , z n ]∗  . . .  v ∗n u = [w 1 , . . . , w n ][µi ][w 1 , . . . , w n ]∗ u.

Here, since [z 1 , . . . , z n ] is an n × n unitary matrix we have that w1 , . . . , w n as given by [w 1 , . . . , w n ] = [v 1 , . . . , v n ][z 1 , . . . , z n ] is an orthonormal basis for N . Setting w n+k = v n+k for k = 1, . . . , m we have that ′

u =

n X k=1

w k µk (u · w k ) +

m X

γk+n wk+n .

k=1

The importance of this mapping is that u′ · wk = µk (u · wk ) for all k = 1, . . . , n and u′ · wk = γk for all k = n + 1, . . . , 2K where u · wk =



u · w k , k = 1, . . . , n 0, k = n + 1, . . . , 2K.

Hence ∆ and Γ may be written as # " wn w1 ,..., p , 0, . . . , 0 ∆= p 1 + |µ1 |2 1 + |µn |2 and

"

Γ= p

µ1 w 1 1 + |µ1 |2

,..., p

µn w n 1 + |µn |2

#

, w n+1 , . . . , w 2K .

It should be noted that the identities ∆∗ ∆ + Γ∗ Γ = I and Γ∗ ∆ = ∆∗ Γ still hold. ˜ and ∆ ˜ can be written in the Similarly we may assume, without loss of generality, that Γ form of Theorem 6.1.

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