An inverse problem for the non-selfadjoint matrix

c de Gruyter 2007

DOI 10.1515 / JIIP.2007.002

J. Inv. Ill-Posed Problems 15 (2007), 1–14

An inverse problem for the non-selfadjoint matrix Sturm–Liouville equation on the half-line G. Freiling and V. Yurko

Abstract. An inverse spectral problem is studied for the non-selfadjoint matrix Sturm–Liouville differential equation on the half-line. We give a formulation of the inverse problem, prove the corresponding uniqueness theorem and provide a constructive procedure for the solution of the inverse problem by the method of spectral mappings. The obtained results are natural generalizations of the classical results in inverse problem theory for scalar Sturm-Liouville operators. Key words. Matrix Sturm-Liouville operators, inverse spectral problems, Weyl matrix, method of spectral mappings. AMS classification. 34A55 34B24 34L40 47E05.

1. Introduction Consider the matrix Sturm–Liouville equation `Y := −Y 00 + Q(x)Y = λY,

x > 0,

(1.1)

with the boundary condition U (Y ) := HY 0 (0) − hY (0) = 0.

(1.2)

Here Y = [yk ]k=1,m is a column-vector, λ is the spectral parameter, h = [hkν ]k,ν =1,m , H = [Hkν ]k,ν =1,m , and Q(x) = [Qkν (x)]k,ν =1,m are matrices, where Qkν (x) are integrable complex-valued functions, Hkν , hkν are complex numbers, and rank [H, h] = m. The matrix Q(x) is called the potential. We study the inverse problem of spectral analysis for the non-selfadjoint matrix Sturm-Liouville boundary value problem of the form (1.1), (1.2). Inverse spectral problems consist in recovering operators from their spectral characteristics. Such problems often appear in many branches of natural sciences and engineering. The scalar case (m = 1) has been studied fairly completely (see the monographs [9, 12–14] and the references therein). The investigation of the matrix case is more difficult, and nowadays there are only isolated fragments, not constituting a general picture, in the inverse problem theory for matrix Sturm–Liouville differential operators. Some aspects of the inverse problem theory for the matrix case were studied in [1,2,4– 7, 10, 11, 15, 18, 23] and other works, but only particular questions and mostly for the selfadjoint case are considered there. We note that inverse problems for various classes of first order differential systems were studied in [3, 16, 17, 19, 22, 24] and other works.

2

G. Freiling and V. Yurko

In this paper we investigate an inverse problem for matrix Sturm–Liouville operators on the half-line which is a natural generalization of the well-known inverse problem for the classical scalar Sturm-Liouville operators. Note that we consider not only selfadjoint case but also the non-selfadjoint one. As a main spectral characteristic we use the so-called Weyl matrix, which is a generalization of the Weyl function (m-function) for the scalar case (see [9, 13]). The Weyl functions and their generalizations often appear in applications and in pure mathematical problems, and they are the most natural spectral characteristics in the inverse problem theory for various classes of differential operators. In this paper for studying the inverse problem for matrix Sturm–Liouville operators we develop the ideas of the method of spectral mappings [21]. In Section 2 the properties of the Weyl matrix are studied. In Section 3 it is proved that the specification of the Weyl matrix uniquely determines the matrix Sturm–Liouville operator. In Section 4 we provide a constructive procedure for the solution of the inverse problem from the Weyl matrix. The central role there is played by the so-called main equation which is a linear integral equation in a corresponding Banach space. We give a derivation of the main equation and prove its unique solvability. Using the solution of the main equation we construct the solution of the inverse problem. We note that the obtained results are nontrivial generalizations of the classical results in the inverse problem theory for scalar Sturm–Liouville operators.

2. The Weyl matrix Let λ = ρ2 , ρ = σ + iτ , and let for definiteness τ := Im ρ ≥ 0. Denote by Π the λ-plane with the cut λ ≥ 0, and Π1 = Π \ {0}; notice that here Π and Π1 must be considered as subsets of the Riemann surface of the square-root-function. Then, under the map ρ → ρ2 = λ, Π1 corresponds to the domain Ω = {ρ: Im ρ ≥ 0, ρ 6= 0}. Put Ωδ = {ρ: Im ρ ≥ 0, |ρ| ≥ δ}. First we construct a special fundamental system of solutions for equation (1.1) in Ω having asymptotic behavior at infinity like exp (±iρx). The following assertions are proved analogously to the scalar case (see [9, Ch. 2]). Lemma 2.1. Equation (1.1) has a unique matrix solution e(x, ρ) = [ekν (x, ρ)]k,ν =1,m , ρ ∈ Ω, x ≥ 0, satisfying the integral equation 1 e(x, ρ) = exp (iρx)Im − 2iρ

Z

∞


exp (iρ(x − t)) − exp (iρ(t − x)) Q(t)e(t, ρ) dt,

x

where Im = [δkν ]k,ν =1,m is the identity matrix, and δkν is the Kronecker symbol. The matrix-function e(x, ρ) has the following properties: (i1 ) For x → ∞, ν = 0, 1, and each fixed δ > 0,
e(ν ) (x, ρ) = (iρ)ν exp (iρx) Im + o(1) ,

uniformly in Ωδ .

(2.1)

3

Sturm–Liouville equation on the half-line

(i2 ) For |ρ| → ∞, ρ ∈ Ω, ν = 0, 1, Z ∞   1  1 e(ν ) (x, ρ) = (iρ)ν exp (iρx) Im − Q(t) dt + o , 2iρ x ρ

(2.2)

uniformly for x ≥ 0. (i3 ) For each fixed x ≥ 0, and ν = 0, 1, the matrix-functions e(ν ) (x, ρ) are analytic for Im ρ > 0, and are continuous for ρ ∈ Ω. The matrix-function e(x, ρ) is called the Jost solution for (1.1). Lemma 2.2. For each δ > 0, there exists a = aδ ≥ 0 such that equation (1) has a unique matrix solution E (x, ρ) = [Ekν (x, ρ)]k,ν =1,m , ρ ∈ Ωδ , satisfying the integral equation E (x, ρ) = exp (−iρx)Im +

1 2iρ

Z

x

exp (iρ(x − t))Q(t)E (t, ρ) dt a

1 + 2iρ

Z

∞

exp (iρ(t − x))Q(t)E (t, ρ) dt. x

The matrix-function E (x, ρ), called the Birkhoff solution for (1.1), has the following properties: (i1 ) E (ν ) (x, ρ) = (−iρ)ν exp (−iρx)(Im + o(1)), x → ∞, ν = 0, 1, uniformly for |ρ| ≥ δ , Im ρ ≥ α, for each fixed α > 0; (i2 ) E (ν ) (x, ρ) = (−iρ)ν exp (−iρx)(Im + O(ρ−1 )), |ρ| → ∞, ρ ∈ Ω, uniformly for x ≥ a; (i3 ) for each fixed x ≥ 0, the matrix-functions E (ν ) (x, ρ) are analytic for Im ρ > 0, |ρ| ≥ δ , and are continuous for ρ ∈ Ωδ ; (i4 ) the matrix-functions e(x, ρ) and E (x, ρ) form a fundamental system of solutions for equation (1.1); Z ∞

(i5 ) if δ ≥

Q(t) dt, then one can take above a = 0. 0

Everywhere below let for definiteness H = Im , i. e. (1.2) takes the form U (Y ) := Y 0 (0) − hY (0) = 0.

(2.3)

Other cases can be treated similarly. Denote by L = L(Q, h) the boundary value problem of the form (1.1), (2.3). Let ϕ(x, λ) = [ϕjk (x, λ)]j,k=1,m and S (x, λ) = [Sjk (x, λ)]j,k=1,m be the solutions of (1.1) under the initial conditions ϕ(0, λ) = Im , ϕ0 (0, λ) = h, S (0, λ) = 0, S 0 (0, λ) = Im , where 0 is the zero m × m matrix. Clearly, U (ϕ) = 0. For each fixed x, the matrix-functions ϕ(ν ) (x, λ) and S (ν ) (x, λ), ν = 0, 1, are entire in λ. The matrix-functions ϕ(x, λ) and S (x, λ) form a fundamental system of solutions for equation (1.1), and det[ϕ(ν ) (x, λ), S (ν ) (x, λ)]ν =0,1 ≡ 1. Denote u(ρ) := U (e(x, ρ)) = e0 (0, ρ) − he(0, ρ),

∆(ρ) := det u(ρ).

4

G. Freiling and V. Yurko

By virtue of Lemma 2.2, the functions u(ρ) and ∆(ρ) are analytic for Im ρ > 0, and continuous for ρ ∈ Ω. It follows from (2.2) that for |ρ| → ∞, ρ ∈ Ω, Z ∞ 1 1 e(0, ρ) = Im − Q(t) dt + o , 2iρ 0 ρ Z ∞   1  1 h u(ρ) = (iρ) Im − Q(t) dt − +o , 2iρ 0 iρ ρ
∆(ρ) = (iρ)m 1 + O(ρ−1 ) .

(2.4)

Denote Λ = {λ = ρ2 : ρ ∈ Ω, ∆(ρ) = 0}, Λ0 = {λ = ρ2 : Im ρ > 0, ∆(ρ) = 0}, Λ00 = {λ = ρ2 : Im ρ = 0, ρ 6= 0, ∆(ρ) = 0}. Obviously, Λ = Λ0 ∪ Λ00 is a bounded set, and Λ0 is a bounded and at most countable set. Denote Φ(x, λ) = e(x, ρ)(u(ρ))−1 . (2.5) The matrix-function Φ(x, λ) = [Φjk (x, λ)]j,k=1,m , x ≥ 0, satisfies (1.1) and on account of Lemma 2.1 also the conditions U (Φ) = Im ,

Φ(x, λ) = O(exp (iρx)),

x → ∞,

ρ ∈ Ω \ Λ.

Denote M (λ) := Φ(0, λ). The matrix M (λ) is called the Weyl matrix for L. It follows from (2.5) that M (λ) = e(0, ρ)(u(ρ))−1 . (2.6) Moreover, Φ(x, λ) = S (x, λ) + ϕ(x, λ)M (λ).

(2.7)

The matrix-functions ϕ(x, λ) and Φ(x, λ) form a fundamental system of solutions for equation (1), and det [ϕ(ν ) (x, λ), Φ(ν ) (x, λ)]ν =0,1 ≡ 1. Theorem 2.3. The Weyl matrix M (λ) is analytic in Π \ Λ0 and continuous in Π1 \ Λ. The set of singularities of M (λ) (as an analytic function) coincides with the set Λ0 := {λ: λ ≥ 0} ∪ Λ. For |ρ| → ∞, ρ ∈ Ω,
M (λ) = (iρ)−1 Im + O(ρ−1 ) . Theorem 2.3 follows from (2.4), (2.6) and Lemma 2.1. Theorem 2.4. The boundary value problem L has no eigenvalues λ > 0.

Sturm–Liouville equation on the half-line

5

Proof. Suppose that λ0 = ρ20 > 0 is an eigenvalue, and let Y0 (x) be a corresponding eigenvector. Since the matrix-functions {e(x, ρ0 ), e(x, −ρ0 )} form a fundamental system of solutions of (1), we have Y0 (x) = e(x, ρ0 )A + e(x, −ρ0 )B , where A and B are constant column-vectors. For x → ∞, Y0 (x) ∼ 0, e(x, ±ρ0 ) ∼ exp (±iρ0 x). But this is possible only if A = B = 0. 2 Theorem 2.5. Let λ0 ∈ / [0, ∞). For λ0 to be an eigenvalue, it is necessary and sufficient that ∆(ρ0 ) = 0. In other words, the set of nonzero eigenvalues coincides with Λ0 . Proof. Let λ0 ∈ Λ0 , i.e. ∆(ρ0 ) = 0. By virtue of (2.1), lim e(x, ρ0 ) = 0. Consider x→∞

the following linear algebraic system u(ρ0 )α = 0 with respect to the column-vector α = [αk ]k=1,m . Since det u(ρ0 ) = ∆(ρ0 ) = 0, this system has a nonzero solution α0 . Put Y0 (x) = e(x, ρ0 )α0 . Then U (Y0 ) = 0, limx→∞ Y0 (x) = 0, and consequently, Y0 (x) is an eigenfunction, and λ0 = ρ20 is an eigenvalue. Conversely, let λ0 = ρ20 , Im ρ0 > 0 be an eigenvalue, and let Y0 (x) be a corresponding eigenfunction. Since the matrix-functions e(x, ρ0 ) and E (x, ρ0 ) form a fundamental system of solutions of equation (1.1), we get Y0 (x) = e(x, ρ0 )a + E (x, ρ0 )b, where a and b are constant column-vectors. As x → ∞, we calculate b = 0, i.e. Y0 (x) = e(x, ρ0 )a, a 6= 0. Since U (Y0 ) = 0, it follows that u(ρ0 )a = 0. Consequently, det u(ρ0 ) = 0, i.e. ∆(ρ0 ) = 0. Theorem 2.5 is proved. 2 The set Λ0 of singularities of the Weyl matrix M (λ) coincides with the spectrum of L. Thus, the spectrum of L consists of the positive half-line {λ : λ ≥ 0}, and the discrete set Λ = Λ0 ∪ Λ00 . Each element of Λ0 is an eigenvalue of L. According to Theorem 2.4, the points of Λ00 are not eigenvalues, they are called spectral singularities of L.

3. The uniqueness theorem The inverse problem is formulated as follows: Given the Weyl matrix M , construct Q and h. In this section we prove the uniqueness theorem for the solution of this inverse problem. For this purpose we agree that together with L we consider a boundary value problem L˜ of the same form but with different Q˜ and h˜ . Everywhere below if a symbol α denotes an object related to L, then α˜ will denote the analogous object related to L˜ , and αˆ = α − α˜ . ˜ , then Q = Q˜ and h = h˜ . Thus, the specification of the Weyl Theorem 3.1. If M = M matrix M uniquely determines the potential Q and the coefficients of the boundary conditions (2.3). Proof. Denote `∗ Z := −Z 00 + ZQ(x),

hZ, Y i := Z 0 Y − ZY 0 ,

U ∗ (Z ) := Z 0 (0) − Z (0)h,

6

G. Freiling and V. Yurko

where Z = [Zk ]tk=1,m is a row vector (t is the sign for the transposition). Then hZ, Y i|x=0 = U ∗ (Z )Y (0) − Z (0)U (Y ).

(3.1)

Moreover, if Y (x, λ) and Z (x, µ) satisfy the equations `Y (x, λ) = λY (x, λ) and `∗ Z (x, µ) = µZ (x, µ), respectively, then d hZ (x, µ), Y (x, λ)i = (λ − µ)Z (x, µ)Y (x, λ). dx

(3.2)

Let ϕ∗ (x, λ), S ∗ (x, λ) and Φ∗ (x, λ) be the matrices satisfying the equation `∗ Z = λZ and the conditions ϕ∗ (0, λ) = Im ,

ϕ∗ 0 (0, λ) = h,

U ∗ (Φ∗ ) = Im ,

Φ∗ (x, λ) = O(exp (iρx)),

S ∗ (0, λ) = 0,

S ∗ 0 (0, λ) = Im , x → ∞,

ρ ∈ Ω.

Denote M ∗ (λ) := Φ∗ (0, λ). Then Φ∗ (x, λ) = S ∗ (x, λ) + M ∗ (λ)ϕ∗ (x, λ).

(3.3)

According to (3.2), hΦ∗ (x, λ), Φ(x, λ)i does not depend on x. Using (3.1) we calculate hΦ∗ (x, λ), Φ(x, λ)i|x=0 = M (λ) − M ∗ (λ).

Moreover, limx→∞ hΦ∗ (x, λ), Φ(x, λ)i = 0, and consequently, M ∗ (λ) = M (λ).

(3.4)

Using (3.2) again, we get ϕ∗ (x, λ)Φ0 (x, λ) − ϕ∗ 0 (x, λ)Φ(x, λ) = Im , ϕ∗ (x, λ)ϕ0 (x, λ) − ϕ∗ 0 (x, λ)ϕ(x, λ) = 0,

(3.5)

Φ∗ 0 (x, λ)ϕ(x, λ) − Φ∗ (x, λ)ϕ0 (x, λ) = Im , Φ∗ (x, λ)Φ0 (x, λ) − Φ∗ 0 (x, λ)Φ(x, λ) = 0, hence, "

ϕ(x, λ) ϕ0 (x, λ)

Φ(x, λ) Φ0 (x, λ)

#−1

"

=

Φ∗ 0 (x, λ) −Φ∗ (x, λ) −ϕ∗ 0 (x, λ) ϕ∗ (x, λ)

# .

(3.6)

Using the fundamental system of solutions {e(x, ρ), E (x, ρ)} and the initial conditions on ϕ(x, λ) we calculate ϕ(x, λ) = e(x, ρ)A1 (ρ) + E (x, ρ)A2 (ρ), |ρ| → ∞,

Aj (ρ) = ρ ∈ Ω,


1 Im + O(ρ−1 ) , 2

7

Sturm–Liouville equation on the half-line

and consequently, for |ρ| → ∞, ρ ∈ Ω, x ≥ 0, ν = 0, 1, we obtain
(iρ)ν
(−iρ)ν exp (−iρx) Im + O(ρ−1 ) + exp (iρx) Im + O(ρ−1 ) . 2 2 (3.7) It follows from (2.2), (2.4) and (2.5) that for |ρ| → ∞, ρ ∈ Ω, x ≥ 0, ν = 0, 1,
Φ(ν ) (x, λ) = (iρ)ν−1 exp (iρx) Im + O(ρ−1 ) . (3.8) ϕ(ν ) (x, λ) =

Similarly, we calculate for |ρ| → ∞, ρ ∈ Ω, x ≥ 0, ν = 0, 1,   (iρ)ν
(−iρ)ν exp (−iρx) Im + O(ρ−1 ) + exp (iρx) Im + O(ρ−1 ) , 2 2
Φ∗(ν ) (x, λ) = (iρ)ν−1 exp (iρx) Im + O(ρ−1 ) . (3.9) In particular, (3.7)–(3.9) yield for ρ ∈ Ω, |ρ| ≥ ρ∗ , x ≥ 0, ν = 0, 1, ϕ∗(ν ) (x, λ) =

kϕ(ν ) (x, λ)k, kϕ∗(ν ) (x, λ)k ≤ C|ρ|ν exp (| Im ρ|x), kΦ(ν ) (x, λ)k, kΦ∗(ν ) (x, λ)k ≤ C|ρ|ν−1 exp (−| Im ρ|x).

Now we consider the block-matrix P (x, λ) = [Pjk (x, λ)]j,k=1,2 defined by " # " # ˜ x, λ) ϕ˜ (x, λ) Φ( ϕ(x, λ) Φ(x, λ) P (x, λ) = . ˜ 0 (x, λ) ϕ˜ 0 (x, λ) Φ ϕ0 (x, λ) Φ0 (x, λ)

(3.10)

(3.11)

Taking (3.6) into account we calculate ˜ ∗0 (x, λ) − Φ(j−1) (x, λ)ϕ˜ ∗0 (x, λ), Pj 1 (x, λ) = ϕ(j−1) (x, λ)Φ ˜ ∗ (x, λ). Pj 2 (x, λ) = Φ(j−1) (x, λ)ϕ˜ ∗ (x, λ) − ϕ(j−1) (x, λ)Φ

(3.12)

It follows from (3.7)-(3.10) and (3.12) that kPjk (x, λ)k ≤ C|ρ|j−k , P1k (x, λ) = δ1k Im + O(ρ−1 ),

ρ ∈ Ω,

|ρ| ≥ ρ∗ ,

|ρ| → ∞,

x ≥ 0,

ρ ∈ Θδ ,

k = 1, 2,

x > 0,

(3.13)

k = 1, 2, (3.14)

where Θδ := {ρ: arg ρ ∈ [δ, π − δ ]}, δ > 0. Using (2.7), (3.3), (3.4) and (3.11) we obtain 0 0 Pj 1 (x, λ) = ϕ(j−1) (x, λ)S˜ ∗ (x, λ) − S (j−1) (x, λ)ϕ˜ ∗ (x, λ)

˜ (λ) − M (λ))ϕ˜ ∗0 (x, λ), + ϕ(j−1) (x, λ)(M Pj 2 (x, λ) = S (j−1) (x, λ)ϕ˜ ∗ (x, λ) − ϕ(j−1) (x, λ)S˜ ∗ (x, λ)

˜ (λ))ϕ˜ ∗ (x, λ). + ϕ(j−1) (x, λ)(M (λ) − M

8

G. Freiling and V. Yurko

˜ (λ), it follows that for each fixed x, the matrix-functions Pjk (x, λ) Since M (λ) ≡ M are entire in λ of order 1/2. Together with (3.13)–(3.14) this yields P11 (x, λ) ≡ Im ,

P12 (x, λ) ≡ 0.

By virtue of (3.11), we have ϕ(x, λ) ≡ ϕ˜ (x, λ), for all x and λ, and consequently, Q(x) = Q˜ (x) a.e. on (0, T ), and h = h˜ . Theorem 3.1 is proved. 2

4. Solution of the inverse problem Let the Weyl-matrix M (λ) of the boundary value problem L = L(Q, h) be given. We ˜ h˜ ) (for example, one can choose an arbitrary model boundary value problem L˜ = L(Q, take Q˜ (x) ≡ 0, h˜ = 0). Denote Z x hϕ∗ (x, µ), ϕ(x, λ)i = ϕ∗ (t, µ)ϕ(t, λ) dt, D(x, λ, µ) = λ−µ 0 Z x ∗ ˜ (x, λ, µ) = hϕ˜ (x, µ), ϕ˜ (x, λ)i = D ϕ˜ ∗ (t, µ)ϕ˜ (t, λ) dt, λ−µ 0 ˆ (µ)D(x, λ, µ), r(x, λ, µ) = M

ˆ (µ)D˜ (x, λ, µ). r˜(x, λ, µ) = M

By the standard way (see [9, Ch. 2]) we get the estimates ˜ (x, λ, µ)k ≤ kD(x, λ, µ)k, kD λ = ρ2 ,

µ = θ2 ≥ 0,

Cx exp (| Im ρ|x) , |ρ ∓ θ| + 1

(4.1)

±θ Re ρ ≥ 0.

It follows from (2.4) and (2.6) that ˆ (λ) = O(ρ−2 ), M

|ρ| → ∞,

ρ ∈ Ω.

(4.2)

In the λ-plane we consider the contour γ = γ 0 ∪ γ 00 (with counterclockwise circuit), ˜ ∪ {0}, and γ 00 is the where γ 0 is a bounded closed contour encircling the set Λ ∪ Λ 0 two-sided cut along the arc {λ: λ > 0, λ ∈ / int γ } (see Figure 1).

g

g g

Figure 1.

9

Sturm–Liouville equation on the half-line

Theorem 4.1. The following relation holds ϕ˜ (x, λ) = ϕ(x, λ) +

1 2πi

Z ϕ(x, µ)r˜(x, λ, µ) dµ.

(4.3)

γ

Proof. It follows from (3.10) and (4.1), (4.2) that the integrals in (4.3) converges absolutely and uniformly on γ , for each fixed x ≥ 0. Denote Jγ = {λ : λ ∈ / γ ∪ int γ 0 }. Consider the contour γR = γ ∩ {λ : |λ| ≤ R} with counterclockwise circuit, and also 0 = γ ∪ {λ : |λ| = R} with clockwise circuit. By Cauchy‘s consider the contour γR R integral formula [8, p. 84], Z P1k (x, µ) − δ1k Im 1 0 dµ, λ ∈ int γR , k = 1, 2. P1k (x, λ) − δ1k Im = 2πi γR0 λ−µ Using (3.13), (3.14) we get Z lim R→∞

|µ|=R

P1k (x, µ) − δ1k Im dµ = 0, k = 1, 2, λ−µ

and consequently, 1 P1k (x, λ) = δ1k Im + 2πi since

1 2πi

Z γ

Z γ

P1k (x, µ) dµ, λ−µ

dµ = 0, λ−µ

λ ∈ Jγ , k = 1, 2,

(4.4)

λ ∈ Jγ

by Cauchy’s theorem. In (4.4) the integral is understood in the principal value sense: Z Z = lim . γ

R→∞

γR

Since ϕ(x, λ) = P11 (x, λ)ϕ˜ (x, λ) + P12 (x, λ)ϕ˜ 0 (x, λ),

it follows from (4.4) that ϕ(x, λ) = ϕ˜ (x, λ) +

1 2πi

Z γ

P11 (x, µ)ϕ˜ (x, λ) + P12 (x, µ)ϕ˜ 0 (x, λ) dµ, λ−µ

λ ∈ Jγ .

Taking (3.12) into account we get 1 ϕ(x, λ) = ϕ˜ (x, λ) + 2πi

Z 

(ϕ(x, µ)Φ˜∗0 (x, µ) − Φ(x, µ)ϕ˜∗0 (x, µ))ϕ˜ (x, λ)

γ

 ˜ ∗ (x, µ))ϕ˜ 0 (x, λ) dµ . + (Φ(x, µ)ϕ˜ ∗ (x, µ) − ϕ(x, µ)Φ λ−µ

In view of (2.7), (3.3) and (3.4) this yields (4.3), since the terms with S (x, µ) and S ∗ (x, µ) vanish by Cauchy‘s theorem. 2

10

G. Freiling and V. Yurko

For each fixed x ≥ 0, the relation (4.3) can be considered as a linear integral equation with respect to ϕ(x, λ), λ ∈ γ . This equation is called the main equation of the inverse problem. Thus, the nonlinear inverse problem is reduced to the solution of this linear integral equation. Now we are going to prove the unique solvability of the main equation. For this purpose we need the following assertion. Lemma 4.2. The following relation holds r˜(x, λ, µ) − r(x, λ, µ) −

1 2πi

Z r(x, ξ, µ)r˜(x, λ, ξ ) dξ = 0.

(4.5)

γ

Proof. Since 1  1 1  1 − = , λ−µ λ−ξ µ−ξ (λ − ξ )(ξ − µ) we have by Cauchy‘s integral formula Z 1 P (x, ξ ) P (x, λ) − P (x, µ) = dξ, λ−µ 2πi γR0 (λ − ξ )(ξ − µ)

0 λ, µ ∈ int γR .

Using (3.13) we get Z

lim

R→∞

|ξ|=R

P (x, ξ ) dξ = 0, (λ − ξ )(ξ − µ)

and consequently, 1 P (x, λ) − P (x, µ) = λ−µ 2πi

Z γ

P (x, ξ ) dξ, (λ − ξ )(ξ − µ)

λ, µ ∈ Jγ .

(4.6)

It follows from (4.6) that " # ϕ ˜ ( x, λ ) P ( x, λ ) − P ( x, µ ) [ϕ∗ (x, µ), −ϕ∗ (x, µ)] λ−µ ϕ˜ 0 (x, λ) " # Z 0 ϕ ˜ ( x, λ ) 1 dξ = [ϕ∗ (x, µ), −ϕ∗ (x, µ)]P (x, ξ ) , 0 2πi γ ( λ − ξ )(ξ − µ) ϕ˜ (x, λ) 0

λ, µ ∈ Jγ .

(4.7) In view of (3.11) we infer "

# ϕ˜ (x, λ) [ϕ (x, µ), −ϕ (x, µ)]P (x, λ) ϕ˜ 0 (x, λ) " # ϕ(x, λ) ∗0 ∗ = [ϕ (x, µ), −ϕ (x, µ)] = hϕ∗ (x, µ), ϕ(x, λ)i. (4.8) ϕ0 (x, λ) ∗0

∗

11

Sturm–Liouville equation on the half-line

Taking (3.12) into account we calculate 0

0

[ϕ∗ (x, µ), −ϕ∗ (x, µ)]P (x, µ) = [ϕ∗ (x, µ), −ϕ∗ (x, µ)]× "

˜ ∗0 (x, µ) − Φ(x, µ)ϕ˜ ∗0 (x, µ), ˜ ∗ (x, µ) ϕ(x, µ)Φ Φ(x, µ)ϕ˜ ∗ (x, µ) − ϕ(x, µ)Φ 0 0 ˜ ∗ (x, µ) − Φ0 (x, µ)ϕ˜ ∗ (x, µ), Φ0 (x, µ)ϕ˜ ∗ (x, µ) − ϕ0 (x, µ)Φ ˜ ∗ (x, µ) ϕ0 (x, µ)Φ

# .

According to (3.5) this yields 0

0

[ϕ∗ (x, µ), −ϕ∗ (x, µ)]P (x, µ) = [ϕ˜ ∗ (x, µ), −ϕ˜ ∗ (x, µ)], and consequently, "

# ϕ˜ (x, λ) [ϕ (x, µ), −ϕ (x, µ)]P (x, µ) ϕ˜ 0 (x, λ) " # ϕ˜ (x, λ) ∗0 ∗ = [ϕ˜ (x, µ), −ϕ˜ (x, µ)] = hϕ˜ ∗ (x, µ), ϕ˜ (x, λ)i. (4.9) ϕ˜ 0 (x, λ) ∗0

∗

Using (3.12), (2.7), (3.3) and (3.4) we get Z 1 dξ P (x, ξ ) 2πi γ (λ − ξ )(ξ − µ) # Z " ˆ (ξ )ϕ˜ ∗0 (x, ξ ) ϕ(x, ξ )M ˆ (ξ )ϕ˜ ∗ (x, µ) −ϕ(x, ξ )M 1 dξ = , ˆ (ξ )ϕ˜ ∗0 (x, µ) ϕ0 (x, ξ )M ˆ (ξ )ϕ˜ ∗ (x, µ) (λ − ξ )(ξ − µ) 2πi γ −ϕ0 (x, ξ )M since the terms with S (x, ξ ) vanish by Cauchy‘s theorem. Therefore " # Z 0 ϕ ˜ ( x, λ ) 1 dξ [ϕ∗ (x, µ), −ϕ∗ (x, µ)]P (x, ξ ) 0 2πi γ ( λ − ξ )(ξ − µ) ϕ˜ (x, λ) Z 1 hϕ∗ (x, µ), ϕ(x, ξ )i ˆ hϕ˜ ∗ (x, ξ ), ϕ˜ (x, λ)i =− M (ξ ) dξ, λ, µ ∈ Jγ . (4.10) 2πi γ ξ−µ λ−ξ Substituting (4.8)–(4.10) into (4.7) we obtain hϕ˜ ∗ (x, µ), ϕ˜ (x, λ)i hϕ∗ (x, µ), ϕ(x, λ)i − λ−µ λ−µ Z 1 hϕ∗ (x, µ), ϕ(x, ξ )i ˆ hϕ˜ ∗ (x, ξ ), ϕ˜ (x, λ)i − M (ξ ) dξ = 0, 2πi γ ξ−µ λ−ξ

or, which is the same ˜ (x, λ, µ) − D(x, λ, µ) − 1 D 2πi

Z

ˆ (ξ )D˜ (x, λ, ξ ) dξ = 0. D(x, ξ, µ)M

γ

ˆ (µ), we arrive at (4.5). Lemma 4.2 is proved. Multiplying this relation by M

2

12

G. Freiling and V. Yurko

Let us consider the Banach space B of continuous bounded on γ matrix-functions z (λ) = [zjk (λ)]j,k=1,m , λ ∈ γ , with the norm kzkB = sup max |zjk (λ)|. λ∈γ j,k=1,m

Theorem 4.3. For each fixed x ≥ 0, the main equation (4.3) has a unique solution ϕ(x, λ) ∈ B . Proof. For a fixed x ≥ 0, we consider the following linear bounded operators in B : Z 1 ˜ Az (λ) = z (λ) + z (µ)r˜(x, λ, µ) dµ, 2πi γ 1 Az (λ) = z (λ) − 2πi

Z z (µ)r(x, λ, µ)z (µ) dµ. γ

Then ˜ (λ) = z (λ) + 1 AAz 2πi

Z

 z (µ) r˜(x, λ, µ) − r(x, λ, µ)

γ

1 − 2πi

Z

 r(x, ξ, µ)r˜(x, λ, ξ ) dξ dµ.

γ

By virtue of (4.5) this yields ˜ (λ) = z (λ), AAz

z (λ) ∈ B.

˜ (λ) = z (λ). Thus, AA ˜ = Interchanging places for L and L˜ , we obtain analogously AAz AA˜ = E , where E is the identity operator. Hence the operator A˜ has a bounded inverse operator, and the main equation (4.3) is uniquely solvable for each fixed x ≥ 0. 2 Using the solution of the main equation one can construct the potential matrix Q(x) and the coefficients of the boundary conditions h. Thus, we obtain the following algorithm for the solution of the inverse problem. Algorithm. Let the matrix-function M (λ) be given. Then 1. Choose L˜ and calculate ϕ˜ (x, λ) and r˜(x, λ, µ). 2. Find ϕ(x, λ) by solving equation (4.3). 3. Construct Q(x) and h via Q(x) = ϕ00 (x, λ)(ϕ(x, λ))−1 − λIm , h = ϕ0 (0, λ). Remark 4.4. Using the main equation one can also obtain necessary and sufficient conditions for the solvability of the inverse problem (see [9, sec. 2.2]). Remark 4.5. For the case of a locally integrable complex-valued potential matrix Q(x) the generalized Weyl-matrix can be introduced as a main spectral characteristic, and the inverse problem can be treated analogously to the scalar case (see [9, sec. 2.5]).

Sturm–Liouville equation on the half-line

13

Remark 4.6. Similar results are obtained for the matrix Sturm–Liouville equation (1.1) on a finite interval. For this purpose the method of spectral mappings (see [20, 21]) has been used.

Acknowledgments. This research was supported in part by Grant 04-01-00007 of the Russian Foundation for Basic Research and by DAAD.

References 1. Z. S. Agranovich and V. A. Marchenko, The Inverse Problem of Scattering Theory. Gordon and Breach, New York, 1963. 2. E. Andersson, On the M -function and Borg-Marchenko theorems for vector-valued SturmLiouville equations. J. Math. Phys. 44, 12 (2003), 6077–6100. 3. R. Beals and R. R. Coifman, Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37 (1984), 39–90. 4. R. Carlson, An inverse problem for the matrix Schr¨odinger equation. J. Math. Anal. Appl. 267 (2002), 564–575. 5. N. K. Chakravarty, A necessary and sufficient condition for the existence of the spectral matrix of a differential system. Indian J. Pure Appl. Math. 25, 4 (1994), 365–380. 6. N. K. Chakravarty and S. K. Acharyya, On an inverse problem involving a second-order differential system. J. Indian Inst. Sci. 71, 3 (1991), 239–258. 7. S. Clark, F. Gesztesy, H. Holden, and B. M. Levitan, Borg-type theorems for matrix-valued Schr¨odinger operators. J. Diff. Equations 167, 1 (2000), 181–210. 8. J. B. Conway, Functions of One Complex Variable. Vol. I. Springer-Verlag, New York, 1995, 2nd ed. 9. G. Freiling and V. A. Yurko, Inverse Sturm–Liouville Problems and their Applications. NOVA Science Publishers, New York, 2001. 10. F. Gesztesy, A. Kiselev, and K. A. Makarov, Uniqueness results for matrix-valued Schr¨odinger, Jacobi, and Dirac-type operators. Math. Nachr. 239/240 (2002), 103–145. 11. B. M. Levitan and M. Jodeit, A characterization of some even vector-valued Sturm-Lioville problems. Mat. Fiz. Anal. Geom. 5, 3–4 (1998), 166–181. 12. B. M. Levitan, Inverse Sturm–Liouville Problems. VNU Sci. Press, Utrecht, 1987. 13. B. M. Levitan and I. S. Sargsjan, Sturm–Liouville and Dirac Operators. Kluwer Academic Publishers, Dordrecht, 1991. 14. V. A. Marchenko, Sturm–Liouville Operators and their Applications. Birkh¨auser, 1986. 15. B. R. Paladhi, The inverse problem associated with a pair of second-order differential equations. Proc. London Math. Soc. 3 43, 1 (1981), 169–192. 16. L. A. Sakhnovich, Spectral Theory of Canonical Differential Systems. Method of Operator Identities. Birkh¨auser Verlag, Basel, 1999. 17. A. B. Shabat, An inverse scattering problem. Differ. Equations 15, 10 (1979), 1299–1307. 18. Shen and Chao-Liang, Some inverse spectral problems for vectorial Hill’s equations. Inverse Problems 17 (2001), 1253–1294.

14

G. Freiling and V. Yurko

19. M. Yamamoto, Inverse spectral problem for systems of ordinary differential equations of first order. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 35, 3 (1988), 519–546. 20. V. A. Yurko, Inverse Spectral Problems for Differential Operators and their Applications. Gordon and Breach, Amsterdam, 2000. 21. 22.

, Method of Spectral Mappings in the Inverse Problem Theory. VSP, Utrecht, 2002. , An inverse spectral problem for singular non-selfadjoint differential systems. Sbornik: Mathematics 195, 12 (2004), 1823–1854.

23.

, Inverse problems for matrix Sturm–Liouville operators. Russian J. Math. Phys. 13, 1 (2006).

24. X. Zhou, Direct and inverse scattering transforms with arbitrary spectral singularities. Comm. Pure Appl. Math. 42 (1989), 895–938.

Received May 30, 2007 Author information G. Freiling, Fachbereich Mathematik, Universit¨at Duisburg-Essen, D-47048 Duisburg, Germany. Email: [email protected] V. Yurko, Department of Mathematics, Saratov State University, Astrakhanskaya 83, 410026 Saratov, Russia. Email: [email protected]