Inverse Wave Spectral Problem with Discontinuous Wave Speed

Journal of Mathematical Physics, Analysis, Geometry 2010, v. 6, No. 3, pp. 255–265

Inverse Wave Spectral Problem with Discontinuous Wave Speed R.F. Efendiev Institute of Applied Mathematics, Baku State University 23 Z. Khalilov Str., Baku, AZ1148, Azerbaijan E-mail:[email protected]

H.D. Orudzhev Qafqaz University 16 km Baku–Sumqayit Road, Baku, AZ0101, Azerbaijan E-mail:[email protected]

Received March 16, 2009 The inverse problem for the Sturm–Liouville operator with complex periodic potential and positive discontinuous coefficients on the axis is studied. The main characteristics of the fundamental solutions and the spectrum of the operator are studied. The formulation of the inverse problem and a constructive procedure for its solution are given. The uniqueness theorem of the inverse problem is proven. Key words: spectral singularities, inverse spectral problem, continuous spectrum. Mathematics Subject Classification 2000: 34A36, 34L05, 47A10, 47A70.

Introduction We consider the differential equation −y 00 (x) + q (x) y (x) = λ2 ρ (x) y (x)

(1)

in the space L2 (−∞, +∞) , where the prime denotes the derivative with respect to the space coordinate. We assume that the potential q (x) is of the form q(x) =

∞ X n=1

This work is supported by PICS-4962. c R.F. Efendiev and H.D. Orudzhev, 2010 °

qn einx ,

(2)

R.F. Efendiev and H.D. Orudzhev the condition

P∞

2 n=1 |qn |

= q < ∞ is satisfied, λ is a complex number, and ½ 1 for x ≥ 0, ρ (x) = 2 β for x < 0, β 6= 1, β > 0 .

(3)

In the frequency domain this equation describes the wave propagation in an inhomogeneous medium, where q(x) is the restoring force and 1/ρ (x) is the wave speed. The discontinuity in ρ (x) usually expresses an abrupt change in the propagation medium of a wave, for example, the tension or the mass density of string, or the refractive index in a wave propagation medium. In regard to the problems with discontinuous coefficients, we remark that Sabatier and his co-writers [1–4] studied the scattering for the impedance-potential equation and the similar problems were intensively studied by many authors in different statements [5, 6], but for the periodic complex potential they are considered for the first time. Firstly potential (2) was considered by M.G. Gasymov [7]. Later, in 1990, the results obtained in [7] were extended by L.A. Pastur, V.A. Tkachenko [8]. As a final remark we mention some works of V. Guillemin, A. Uribe [9] and [10–12]. In the paper, our primary aim is to study the spectrum and to solve the inverse problem for singular nonself-adjoint operator by transmitting the coefficient and normalizing the numbers corresponding to quasieigenfunctions of the Sturm–Liouville operator with complex periodic potential and positive discontinuous coefficients on the axis. As the coefficient allows a bounded analytic continuation to the upper half-plane of the complex plane z = x + it, we can analyze the problem (1)–(3) in detail. The paper consists of three sections. In Section 1 we study the properties of fundamental system of solutions of equation (1). The spectrum of problem (1)–(3) is studied in Section 2. In Section 3 we give a formulation of the inverse problem, prove the uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.

1.

Representation of Fundamental Solutions

Here we study the solutions of the main equation −y 00 (x) + q (x) y (x) = λ2 ρ (x) y (x) that will be needed later. P 2 We can prove the existence of these solutions if the condition ∞ n=1 |qn | = q < ∞ is fulfilled for the potential. This will be unique restriction on the potential and later on we will consider it to be fulfilled.

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Journal of Mathematical Physics, Analysis, Geometry, 2010, v. 6, No. 3

Inverse Wave Spectral Problem with Discontinuous Wave Speed Theorem 1. Let q(x) be of the form (2) and ρ (x) satisfy condition (3). Then equation (1) has special solutions of the form Ã ! ∞ ∞ X X 1 + iλx iαx f1 (x, λ) = e 1+ Vnα e f or x ≥ 0, (4) n + 2λ α=n n=1 Ã ! ∞ ∞ X X 1 + −iλβx iαx f2 (x, λ) = e 1+ Vnα e f or x < 0, (5) n − 2λβ α=n n=1

where the numbers Vnα are determined from the following recurrent relations: α(α − n)Vnα +

α−1 X

qα−s Vns = 0,

1 ≤ n < α,

(6)

s=n

α

α X

Vnα + qα = 0,

(7)

n=1

and the series ∞ ∞ X 1X α |Vnα | n α=n

(8)

n=1

converge. The proof of the theorem is similar to that of [8] and therefore we do not cite it here. n R e m a r k 1. If λ 6= − n2 , λ 6= 2β and Imλ > 0, then f1+ (x, λ) ∈ L2 (0, +∞), f2+ (x, λ) ∈ L2 (−∞, 0). By analogy to [7], it is easy to see that equation (1) has fundamental solutions + f1 (x, λ),f1− (x, λ) (f2+ (x, λ),f2− (x, λ)) for which

£ ¤ W f1+ (x, λ), f1− (x, λ) = 2iλ, £ ¤ W f2+ (x, λ), f2− (x, λ) = −2iλβ, (where W [f, g] = f 0 g − f g 0 ) is satisfied Then each solution of equation (1) may be represented as a linear combination of these solutions f2+ (x, λ) = C11 (λ) f1+ (x, λ) + C12 (λ) f1− (x, λ) f or x ≥ 0,

(9)

f1+ (x, λ) = C22 (λ) f2+ (x, λ) + C21 (λ) f2− (x, λ) f or x < 0,

(10)

where


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R.F. Efendiev and H.D. Orudzhev

− + f1,2 (x, λ) = f1,2 (x, −λ) ,

C11 (λ) =

W [f2+ (0, λ) , f1− (0, λ)] , 2iλ

C12 (λ) =

W [f1+ (0, λ) , f2+ (0, λ)] , 2iλ

(11)

1 1 C22 (λ) = − C11 (−λ) , C21 (λ) = C12 (λ) . β β

(12)

According to physical sense of the solutions f1± (x, λ),f2± (x, λ) , it is natural to 11 (λ) say that C121(λ) is a transmission coefficient to the left and C C12 (λ) is a reflection coefficient from the left for equation (1). Let fn± (x) = limn (n ± 2λ)f1± (x, λ) =

∞ X

n

Vnα eiαx e−i 2 x .

(13)

It follows from relation (6) that¡ fn± (x)¢ 6≡ 0 is valid for Vnn 6= we obtained that¢W [fn± (x) , f1∓ x, ∓ n2 ] = 0, and consequently ¡ ∓ fn± (x) , f1 x, ∓ n2 are linear dependent.

0. From this the functions

λ→∓ 2

α=n

Therefore,

³ n´ fn± (x) = Vnn f1∓ x, ∓ . 2

2.

(14)

Spectrum of Operator L

Let L be an operator generated by the differential expression in the space L2 (−∞, +∞, ρ (x)). Divide the plane λ into sectors Sk = {kπ < arg λ < (k + 1)π},

1 ρ(x)

n o d2 − dx 2 + q (x)

k = 0, 1.

¡ ¢ By means of general method for the kernel of the resolvent of operator L − λ2 I we get ½ + 1 f1 (x, λ) f2+ (t, λ) f or t < x λ ∈ S0 (15) R11 (x, t, λ) = f or t > x C12 (λ) f1+ (t, λ) f2+ (x, λ) and

258


Inverse Wave Spectral Problem with Discontinuous Wave Speed

R12 (x, t, λ) =

1 C12 (−λ)

½

f1− (x, λ) f2− (t, λ) f1− (t, λ) f2− (x, λ)

f or t < x λ ∈ S1 . f or t > x

(16)

R e m a r k 2. Rλ2 = (L − λ2 I)−1 exists and is bounded for all λ2 out of positive half-line and C12 (±λ) 6= 0. Lemma 2. The coefficient C12 (λ) is an analytic function in the Imλ > 0 and has a finite number of zeros, moreover, if C12 (λn ) = 0, then d C12 (λ)|λ=λn = −i dλ

+∞ Z ρ (x) f1+ (x, λn ) f2+ (x, λn ) dx.

−∞

The proof of Lemma 2 is similar to that of [14, p. 173] and therefore we do not cite it here. For the solutions f1± (x, λ) and f2± (x, λ) we can obtain the asymptotic equalities: ±(j)

f1

(0, λ) = ± (iλ)j + o(1)

±(j) f2 (0, λ)

for |λ| → ∞, j = 0, 1,

= ∓ (iλβ)j + o (1)

for |λ| → ∞, j = 0, 1.

For simplicity we prove the first equality. Since f1± (0, λ) = 1 +

∞ X ∞ X Vnα , n ± 2λ α=n

n=1

then ∞ X ∞ ∞ X ∞ X X ¯ ± ¯ |Vnα | |Vnα | ¯f (0, λ)¯ ≤ 1 + q ≤ 1+ 1 |n + 2λ| n=1 α=n n=1 α=n (n + 2Reλ)2 + 4Im2 λ ∞

∞

1 X X α |Vnα | . ≤1+ |Imλ| n α=n n=1

f1± (0, λ)

Therefore, as |λ| → ∞, we obtain = 1 + o (1). Analogously, we can prove the rest of asymptotic equalities as |λ| → ∞ for the solutions f2± (x, λ). Then for the coefficients C12 (λ), C12 (−λ), we get the following asymptotic equalities:


259


C12 (λ) =

1 β+1 (iλβ + iλ) + o (1) = + o (1) , 2iλ 2

(17)

β+1 + o (1) . 2 These asymptotic equalities and analytical properties of the coefficients C12 (λ), C12 (−λ) make valid the following statement. C12 (−λ) =

Lemma 3. The eigenvalues of the operator L are finite and coincide with the square of zeros of the functions C12 (λ), C12 (−λ) from the sectors Sk , k = 0, 1, respectively. R e m a r k 3. Taking into account (17), we can obtain the useful relation β=2

lim

Imλ→∞

C12 (λ) − 1.

(18)

Theorem 2. The continuous spectrum of the operator L fills out the semi axis [0, ∞) and on the continuous spectrum may be spectral singularities at the ³ ´2 ¡ ¢2 n points of the form n2 and 2β , n ∈ N. P r o o f. First, we will prove that the operator L has no positive eigenvalues. We recall that equation (1) has fundamental solutions f1+ (x, λ) , f1− (x, λ) . Then for the case λ2 > 0 the solution of equation (1) can be written in the form µ ¶ ∞ ∞ P P 1 iαx y(x, λ) = C1 ei|λ|x 1 + V e nα n+2|λ| α=n ¶ n=1 µ ∞ ∞ P 1 P +C2 e−i|λ|x 1 + Vnα eiαx . n−2|λ| n=1

α=n

So, y (x, λ) ∈ / L2 (−∞, +∞) since the principle parts of the solutions are periodic. Taking into account Remark 2, we will study the function ½ R11 (x, t, λ), λ ∈ S0 R(x, t, λ) = R12 (x, t, λ), λ ∈ S1 in the neighborhood of poles λ0 from [0, ∞). Then the number λ0 coincides n with one of the numbers 2α , n, α ∈ N . From (15)–(16) it follows that the limit lim (λ − λ0 )R (x, t, λ) = R0 (x, t) exists and R0 (x, t) is a bounded function with

λ→λ0

respect to all the variables. Let θ(x) be an arbitrary finite function. Then ϕ (x) = +∞ R R0 (x, t)θ (t) dt is a bounded solution of equation (1) for λ = λ0 . Therefore,

−∞

260


Inverse Wave Spectral Problem with Discontinuous Wave Speed ϕ (x) = C0 f1+ (x, λ0 ). Comparison of the last relation with formulae (15)–(16) shows that if λ0 6= n2 , n ∈ N , then C0 = 0 and so the kernel of the resolvent has removable singularity at the point λ0 . Thus, there is a case λ0 = n2 , n ∈ N , where the kernel of resolvent has the poles of first order. Since f1+ (x, λ0 ) ∈ / L2 (−∞, +∞) , then λ20 = ( n2 )2 is a spectral singularity of the operator L in the sense of M.N. Naimark [13]. (Analogously, we can show that the resolvent has n 2 n the poles of first order at the points λ0 = 2β , n ∈ N and ( 2β ) are spectral singularities of the operator L.) In order all numbers λ2 > 0 belong to the continuous ¡spectra, ¢it suffices to show that the domain of value RL−λ2 I of the operator L − λ2 I is dense in L2 (−∞, +∞), so that the orthogonal complement of the set RL−λ2 I consists of only zero element. However, the orthogonal complement of the set RL−λ2 I coincide with the space of the solutions of equation L∗ f = λ2 f . It is easy to see that the operator L∗ is adjoint to the operator L. Let ψ (x) ∈ L2 (−∞, +∞) , ψ (x) 6= 0, and +∞ Z ¡ ¢ Lf − λ2 f ψ (x)dx = 0

(19)

−∞

be satisfied for any f (x) ∈ D (L). From (19) it follows that ψ (x) ∈ D (L∗ ) , and ψ (x) is an eigenfunction of the operator L∗ corresponding to eigenvalues λ. More exactly, ψ (x) is the solution of the equation −z 00 + q (x) z = λ2 ρ(x)z (20) belonging to L2 (−∞, +∞). We obtained that ψ (x) = 0, since the operator generated by the expression in the left-hand side of (20) is an operator of type ¡ L. This¢ contradiction shows that the domain of value RL−λ2 I of the operator L − λ2 I is everywhere dense in L2 (−∞, +∞). The theorem is proved. Now, taking (9)–(10) and (14) into account, we calculate W [f + ,f − ]

1 [f1+ (x, λ) f1+ (t, λ) W [f2+ ,f1+ ] lim (n − 2λ) R11 (x, t, λ) = lim (n − 2λ) 2iλ 1 2 λ→n/2 λ→n/2 ¡ n¢ + ¡ n¢ + − + 1 +f1 (x, λ) f1 (t, λ)] = in [Vnn f1 x, 2 f1 t, 2 ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ 2 +Vnn f1+ x, n2 f1+ t, n2 ] = in Vnn f1+ x, n2 f1+ t, n2 . (21) Analogously, taking into account that f˜2+ (x, λ) = f2+ (x, λ) (n − 2λβ) has no


261

R.F. Efendiev and H.D. Orudzhev poles at the points λ =

n 2β ,

n ∈ N , we get h ³ ´ ³ í ³ ´ W f2− 0,n/2β ,f1+ 0,n/2β 1 [− in h + ³ n ´ ˜+ ³ n í f˜2+ x, n/2β W f1 0, /2β ,f2 0, /2β

lim (n − 2λβ) R11 (x, t, λ) = λ→n/2β h ³ ´ ³ í ³ ´ ³ ´ W f1+ 0,n/2β ,f˜2+ 0,n/2β − n/ ˜+ t, n/ f x, ] f + 2 2β 2β 2 in

3.

³ ´ = F x, t, n/2β

Eigenfunction Expansions

To define the natural spectral data of the operator L, it is necessary to obtain the eigenfunction expansion for the same operator. For this we consider the following lemma. Lemma 4. Let ψ (x) be an arbitrary twice continuously differentiable function belonging to L2 (−∞, +∞, ρ (x)) . Then +∞ +∞ Z Z 1 ψ (x) R (x, t, λ) g (t) dt, R (x, t, λ) ρ (t) ψ (t) dt = − 2 + 2 λ λ −∞

−∞

where g (t) = −ψ 00 (x) + q (x) ψ (x) ∈ L2 (−∞, +∞) . Integrating the both hand sides along the circle |λ| = R and passing to the limit as R → ∞, we get 1 ψ (x) = − lim R→∞ 2πi

+∞ Z 2λdλ R (x, t, λ) ρ (t) ψ (t) dt.

I |λ|=R

−∞

R +∞ The function −∞ R (x, t, λ) ρ (t) ψ (t) dt is analytical inside the contour with ren spect to λ excepting the points λ = λn , n = 1, 2, . . . , λ = n2 , λ = 2β , n = ¡ ¢ + − 1 1 1, 2, . . . . Denote by Γ0 Γ0 the contour formed by segments [0, 2β − δ], [ 2β + n n 1 δ, 2 − δ], . . . [ 2β + δ, 2 − δ] and semicircles of radius δ with the centers at points n n 2 , 2β n = 1, 2, . . . , located in the upper (lower) half plane. Then 1 ψ (x) = − 2iπ

262

+∞ Z Z Z 2λρ (t) ψ (t) [ R11 (x, t, λ) dλ − R12 (x, t, λ) dλ]dt −∞

Γ+ 0

Γ− 0


Inverse Wave Spectral Problem with Discontinuous Wave Speed

1 =− 2iπ

+∞ Z Z 2λρ (t) ψ (t) [R11 (x, t, λ) − R12 (x, t, λ)]dλdt −∞

Γ− 0

+Res R11 (x, t, λ) +Res R11 (x, t, λ) +Res R11 (x, t, λ) . n n λ=λn

λ= 2β

λ= 2

Calculate separately every term. R11 (x, t, λ) − R12 (x, t, λ) =

f1+ (x, λ) f1+ (t, λ) 2iλC12 (λ) C22 (λ)

The residues of the resolvent R11 (x, t, λ) in λ1, λ2, . . . λl denote by G11 (λn , x, t). Thus G11 (λn , x, t) will be equal to G11 (λn , x, t) = lim (λ − λn ) R11 (x, t, λ) . λ→λn

Then for every function ψ (x) belonging to L2 (−∞, +∞, ρ (x)) we get the eigenfunction expansion in the form 1 ψ (x) = − 2iπ

+∞ R −∞

ρ (t) ψ (t)

hH Γ−

f1+ (x,λ)f1+ (t,λ) 2iλC12 (λ)C22 (λ) dλ

³ í ¡ 0 ¢ ¡ ¢ 2 +G11 (λn , x, t) + in Vnn f1+ x, n2 f1+ t, n2 + F x, t, n/2β dt

4.

(22)

Solution of the Inverse Problem

Let us study the inverse problem for the problem (1–3). From the representation (15)–(16) it also follows that for each x and t from (−∞, +∞) the kernel R(x, t, λ) admits a meromorphic continuation from the sector S = {λ : ³ ´2 ¡ ¢2 n 0 < argλ < π} and may have poles at the points n2 and 2β , n ∈ N , outside of S. These poles of the resolvent are called quasi-stationary states of the operator L. ¡ ¢2 Thus the quasistationary states of the operator L are the numbers n2 and ³ ´2 n , n ∈ N . In spectral expansion (22) the numbers Vnn , n ∈ N , play a part of 2β the normalizing numbers corresponding to quasieigenfunction of the operator L. So, it makes natural the formulation of the inverse problem about reconstruction of the potential of the equation (1) and the number β.


263


Inverse Problem Given the spectral data { C12 (λ) , Vnn }, construct β and the potential q (x). Using the results obtained above, we arrive at the following procedure for the solution of the inverse problem: 1. Taking into account (14), we get Vn,α+n = Vnn

α X Vmα , m+n

m=1

from which all the numbers Vnα , α = 1, 2, . . . , n = 1, 2, . . . , n < α, are defined. 2. From recurrent formula (6)–(8), find all numbers qn . 3. The number β is defined by the formula β=2

lim

Imλ→∞

C12 (λ) − 1.

So, the inverse problem has a unique solution and the numbers β and qn are defined constructively by spectral data. Theorem 3. The specification of spectral data uniquely determines β and the potential q (x). Acknowledgements. The paper was finished during the time R.F. Efendiev visited Universite de Nantes as an invited professor. The author wishes to thank Prof. A. Nachaoui and Prof. A. Boulkhemair from Nantes University for discussing the paper as well as the administration of University for invitation and Laboratoire de Mathematiques Jean Leray for kind and warm hospitality. References [1] P.C. Sabatier and B. Dolveck-Guilpard, On Modelling Discontinuous Media. OneDimensional Approximations. — J. Math. Phys. 29 (1998), 861–868. [2] P.C. Sabatier, On Modelling Discontinuous Media. Three-Dimensional Scattering. — J. Math. Phys. 30 (1989), 2585–2598. [3] F. Dupuy and P.C. Sabatier, Discontinuous Media and Undetermined Scattering Problems. — J. Phys. A 25 (1992), 4253–4268. [4] F.R Molino and P.C. Sabatier, Elastic Waves in Discontinuous Media: ThreeDimensional Scattering. — J. Math. Phys. 3 (1994), 4594–4635. [5] T. Aktosun, M. Klaus, and C. van der Mee, On the Riemann–Hilbert Problem for the One-Dimensional Schrodinger Equation. — J. Math. Phys. 34 (1993), 2651– 2690. [6] I.M. Guseinov and R.T. Pashaev, On an Inverse Problem for a Scond-Order Differential Equation. — UMN 57:3 (2002), 147–148.

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Inverse Wave Spectral Problem with Discontinuous Wave Speed [7] M.G. Gasymov, Spectral Analysis of a Class Nonself-Adjoint Operator of the Second Order. — Funct. Anal. and its Appl. 34 (1980), 14–19. (Russian) [8] L.A. Pastur and V.A. Tkachenko, An Inverse Problem for One Class of one Dimensional Schrodinger’s Operators with Complex Periodic Potentials. — MATH USSR IZV 37 (3) (1991), 611–629. [9] V. Guillemin and A. Uribe, Hardy Functions and the Inverse Spectral Method. — Comm. Part. Diff. Eq. 8 (13) (1983), 1455–1474. [10] R.F. Efendiev, Spectral Analysis of a Class of Nonself-Adjoint Differential Operator Pencils with a Generalized Function. — Teor. Math. Fiz. 145 (2005), 102–107. (Russian) (Engl. Transl.: Theoret. Mat. Phys. 145 (2005), 1457–461.) [11] R.F. Efendiev, Complete Solution of an Inverse Problem for one Class of the High Order Ordinary Differential Operators with Periodic Coefficients. — J. Mat. Fiz., Anal., Geom. 2 (2006), 73–86. [12] R.F. Efendiev, The Characterization Problem for One Class of Second Order Operator Pencil with Complex Periodic Coefficients. — Moscow Math. J. 7 (2007), No. 1, 55–65. [13] M.A. Naimark, Linear Differential Operators. Nauka, Moscow, 1969. (Russian) [14] V.A. Marchenko, Sturm–Liouville Operators and Applications. Naukova Dumka, Kiev, 1977. (Russian)


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