a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition

Volume 2010, Article ID 171967,17pages doi:10.1155/2010/171967

Research Article

On an Inverse Scattering Problem for

a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition

Khanlar R. Mamedov

Mathematics Department, Science and Letters Faculty, Mersin University, 33343 Mersin, Turkey

Correspondence should be addressed to Khanlar R. Mamedov,[email protected] Received 9 April 2010; Accepted 22 May 2010

Academic Editor: Michel C. Chipot

Copyrightq2010 Khanlar R. Mamedov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line0,∞with a linear spectral parameter in the boundary condition. The scattering data of the problem are defined and a new fundamental equation is derived, which is different from the classical Marchenko equation. With help of this fundamental equation, in terms of the scattering data, the potential is recovered uniquely.

1. Introduction

We consider inverse scattering problem for the equation

−ψqxψλ²ρxψ 0< x <∞, 1.1

with the boundary condition

−

α₁ψ0−α₂ψ0 λ²

β₁ψ0−β₂ψ0

, 1.2

whereλis a spectral parameter,qxis a real-valued function satisfying the condition _∞

0

1xqxdx <∞, 1.3

ρxis a positive piecewise-constant function with a finite number of points of discontinuity, α_i, β_i i1,2are real numbers, andγα₁β₂−α₂β₁>0.

The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line 0,∞ for the boundary value problem 1.1–1.3. In the case ρx ≡ 1, the inverse problem of scattering theory for1.1with boundary condition not containing spectral parameter was completely solved by Marchenko1,2, Levitan3,4, Aktosun5, as well as Aktosun and Weder6. The discontinuous version was studied by Gasymov7 and Darwish 8. In these papers, solution of inverse scattering problem on the half-line 0,∞by using the transformation operator was reduced to solution of two inverse problems on the intervals 0, a and a,∞. In the caseρx/1, the inverse scattering problem was solved by Guse˘ınov and Pashaev9by using the newnontriangularrepresentation of Jost solution of1.1. It turns out that in this case the discontinuity of the function ρxstrongly influences the structure of representation of the Jost solution and the fundamental equation of the inverse problem. We note that similar cases do not arise for the system of Dirac equations with discontinuous coefficients in10. Uniqueness of the solution of the inverse problem and geophysical application of this problem for1.1whenqx≡0 were given by Tihonov11 and Alimov12. Inverse problem for a wave equation with a piecewise-constant coefficient was solved by Lavrent’ev13. Direct problem of scattering theory for the boundary value problem1.1–1.3in the special case was studied in14.

When ρx ≡ 1 in 1.1 with the spectral parameter appearing in the boundary conditions, the inverse problem on the half-line was considered by Pocheykina-Fedotova15 according to spectral function, by Yurko16–18according to Weyl function, and according to scattering data in19,20. This type of boundary condition arises from a varied assortment of physical problems and other applied problems such as the study of heat conduction by Cohen 21and wave equation by Yurko16, 17. Spectral analysis of the problem on the half-line was studied by Fulton22.

Also, physical application of the problem with the linear spectral parameter appearing in the boundary conditions on the finite interval was given by Fulton23. We recall that inverse spectral problems in finite interval for Sturm-Liouville operators with linear or nonlinear dependence on the spectral parameter in the boundary conditions were studied by Chernozhukova and Freiling24, Chugunova25, Rundell and Sacks26, Guliyev27, and other works cited therein.

This paper is organized as follows. InSection 2, the scattering data for the boundary value problem1.1–1.3are defined. InSection 3, the fundamental equation for the inverse problem is obtained and the continuity of the scattering function is showed. Finally, the uniqueness of solution of the inverse problem is given inSection 4.

For simplicity we assume that in1.1the functionρxhas a discontinuity point:

ρx

⎧⎨

⎩

α², 0≤x < a,

1, x≥a, 1.4

where 0< α /1.

The function

f₀x, λ 1

2 1 1

ρx

e^iλμx1

2 1− 1 ρx

e^iλμ−x, 1.5

is the Jost solution of1.1whenqx≡0,whereμ^±x ±x

ρx a1∓

ρx.

It is well known9that, for allλfrom the closed upper half-plane,1.1has a unique Jost solutionfx, λwhich satisfies the condition

xlim→∞fx, λe^−iλx1 1.6

and it can be represented in the form

fx, λ f₀x, λ _∞

μxKx, te^iλtdt, 1.7

where the kernelKx, tsatisfies the inequality _∞

μx|Kx, t|dt≤C

exp _∞

x

tqtdt

, 0< Cconst, 1.8

and possesses the following properties:

dK

x, μx

dx − 1

4

ρx 1 1

ρx

qx, 1.9

d dx

K

x, μ⁻x 0

−K

x, μ⁻x−0 1

4

ρx 1− 1

ρx

qx. 1.10

In addition, ifqxis differentiable,Kx, tsatisfiesa.e.the equation

ρx∂²K

∂t² −∂²K

∂x² qxK0, 0< x <∞, t > μx. 1.11 Denote that

ϕλ

α₂β₂λ²

f0, λ−

α₁β₁λ²

f0, λ. 1.12

According toLemma 2.2inSection 2, the equationϕλ 0 has only a finite number of simple roots in the half-plane Imλ >0; all these roots lie in the imaginary axis. The behavior of this boundary value problem1.1–1.3is expressed as a self-adjoint eigenvalue problem.

We will call the function

Sλ

α₂β₂λ²

f0, λ−

α₁β₁λ² f0, λ α₂β₂λ²

f0, λ−

α₁β₁λ²

f0, λ 1.13

the scattering function for the boundary value problem1.1–1.3, wheref0, λdenotes the complex conjugate off0, λ.

We denote bym⁻²_k the normalized numbers for the boundary problem1.1–1.3:

m⁻²_k ≡ _∞

0

ρxfx, iλk²dx 1

γβ₂f0, iλk−β₁f0, iλk², 1.14 wherek1,2, . . . , n. It turns out that the potentialqxin the boundary value problem1.1–

1.3is uniquely determined by specifying the set of values{Sλ, λk, mk}.The set of values is called the scattering data of the boundary value problem1.1–1.3. The inverse scattering problem for boundary value problem1.1–1.3consists in recovering the coefficientqx from the scattering data.

The potentialqxis constructed by slightly varying the method of Marchenko. Set

F0x 1 2π

_∞

−∞S0λ−Sλe^−iλxdλⁿ

k1

m²_ke^−λkx,

F x, y

1

2 1 1

ρx

F0

yμx 1

2 1− 1 ρx

F0

yμ⁻x ,

1.15

where

S0λ

⎧⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎩ f00, λ

f₀0, λ e^−2iλa1τe^−2iλaα

e^−2iλaατ , ifβ2 0, f₀0, λ

f₀0, λ −e^−2iλa1−τe^−2iλaα

e^−2iλaα−τ , ifβ2/0,

1.16

andτ α−1/α1.

We can write out the integral equation

F x, y

_∞

μxKx, tF0

ty dtK

x, y

1− ρx

1

ρxK

x,2a−y

0, 1.17

for the unknown function Kx, t. The integral equation is called the fundamental equation of the inverse problem of scattering theory for the boundary problem 1.1–1.3. The fundamental equation is different from the classic equation of Marchenko and we call the equation the modified Marchenko equation. The discontinuity of the function ρx strongly influences the structure of the fundamental equation of the boundary problem1.1–1.3. By Theorem 4.1inSection 4, the integral equation has a unique solution for everyx≥0. Solving this equation, we find the kernelKx, yof the special solution1.7, and hence according to formula1.10it is constructed the potentialqx.

We show that formula1.7is valid for1.1. For this, let us give the algorithm of the proof in9. Forfx, λlet us consider the integral equation

fx, λ f0x, λ _∞

x

Φx, t, λqtft, λdt, 1.18

where

Φx, t, λ s₀t, λc0x, λ−s₀x, λc0t, λ, 1.19

whiles0x, λandc0x, λare solutions of1.1whenqx≡0, satisfying the initial conditions c00, λ s₀0, λ 1 andc₀0, λ s00, λ 0.

It is not hard to show that the functionΦx, t, λsatisfies the formula

Φx, t, λ _σx,t

−σx,tK₀x, t, ze^iλzdz, 1.20

where

K₀x, t, z

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩ 1

2α, |z| ≤σx, t, x≤t≤a, 1

4

1 1 α

, t−a−αa−x≤ |z| ≤σx, t, x≤a≤t, 1

2, |z| ≤t−a−αa−x, x≤a≤t, 1

2, |z| ≤σx, t, t≥x≥a,

σx, t _t

x

ρsds

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

αt−x, x≤t≤a, αa−x t−a, x≤a≤t, t−x, a≤x≤t.

1.21

Substituting the expression1.7forfx, λin the integral equation1.18and using formula 1.20forΦx, t, λafter elementary operations, the following integral equations for the kernel Kx, tare obtained:

Kx, t 1 4α

11

α

a

αxαa−at/2αqzdz 1

4α

1−1 α

a

αxαaa−t/2αqzdz

1 4

1 1

α

∞ a

qzdz−1 4

1− 1

α

t−αxαaa/2 a

qzdz

1 2α

mint,ααa−a/α x

qz

_tαz−x

t−αz−xKz, sds dz

−1 4

_{taαa−αx/2}

a

qz

_{t−z−a−αaαx}

tz−a−αaαxKz, sds dz,

1.22

for 0< x < a,αx−αaa < t <−αxαaa;

Kx, t 1 4

1 1

α

∞

tαx−αaa/2qzdz 1

4

1− 1 α

∞

t−αxαaa/2qzdz

1 2α

_a

x

qz

_tαz−x

t−az−xKz, sds dz

−1 4

1− 1

α

aαa−αx a

qz

_{t−zaαa−αx}

tz−a−αaαxKz, sds dz 1

4

1− 1 α

∞

aαa−αxqz

_{tz−a−αaαx}

t−zaαa−αxKz, sds dz,

1.23

for 0< x < a, t >−αxαaa;

Kx, t 1 2

_∞

xt/2qzdz 1

2 _∞

x

qzdz _tz−x

t−z−xKz, sds, 1.24

fort≥x≥a.

The solvability of these integral equations is obtained through the method of successive approximations. By using integral equations 1.22–1.24 forKx, t,equalities 1.9, 1.10 are obtained. By substituting the expressions for the functions fx, λ and fx, λin1.1, it can be shown that1.11holds.

2. The Scattering Data

For real λ /0, the functionsfx, λ andfx, λform a fundamental system of solutions of 1.1and their Wronskian is computed as W{fx, λ, fx, λ} 2iλ. Here the Wronskian is defined asW{f, g}fg−fg.

Letωx, λbe the solution of1.1satisfying the initial condition

ω0, λ α₂β₂λ², ω0, λ α₁β₁λ². 2.1

The following assertion is valid.

Lemma 2.1. The identity

2iλωx, λ α2β2λ²

f0, λ−

α1β1λ²

f0, λ fx, λ−Sλfx, λ 2.2

holds for all realλ /0, where

Sλ

α2β2λ²

f0, λ−

α1β1λ² f0, λ α2β2λ²

f0, λ−

α1β1λ²

f0, λ 2.3

with

Sλ S−λ S−λ⁻¹. 2.4

The functionSλis called the scattering function of the boundary value problem1.1–

1.3.

Lemma 2.2. The functionϕλmay have only a finite number of zeros in the half-plane Imλ > 0.

Moreover, all these zeros are simple and lie in the imaginary axis.

Proof. Sinceϕλ/0 for all realλ /0, the pointλ 0 is the possible real zero of the function ϕλ. Using the analyticity of the function ϕλ in upper half-plane and the properties of solution1.7are obtained that zeros ofϕλform at most countable and bounded set having 0 as the only possible limit point.

Now let us show that all zeros of the functionϕλlie on the imaginary axis. Suppose thatμ1andμ2are arbitrary zeros of the functionϕλ. We consider the following relations:

−f x, μ₁

qxf x, μ₁

μ²₁ρxf x, μ₁

,

−f x, μ2

qxf x, μ2

μ22ρxf x, μ2

.

2.5

Multiplying the first of these relations byfx, μ2and the second byfx, μ1, subtracting the second resulting relation from the first, and integrating the resulting difference from zero to infinity, we obtain

μ²₁−μ22 ∞ 0

ρxf x, μ1

f x, μ2

dx−W f

x, μ1

, f x, μ2

x00. 2.6

On the other hand, according to the definition of the functionϕλ, the following relation holds:

ϕ μ_j

α₂β₂μ²_j f

0, μ_j

−

α₁β₁μ²_j f

0, μ_j

0, j1,2. 2.7

Therefore,

f x, μ_j

1 γ

β₂f 0, μ_j

−β₁f 0, μ_j

ω x, μ_j

, j1,2. 2.8

This formula yields

W f

x, μ1

, f x, μ2

x0 1

γ β2f

0, μ1

−β1f 0, μ1

× β₂f

0, μ₂

−β₁f

0, μ₂

μ₂²−μ²₁ .

2.9

Thus, using2.6and2.9we have

μ²₁−μ22^∞

0

ρxf x, μ1

f x, μ2

dx1 γ

β2f 0, μ1

−β1f 0, μ1

× β₂f

0, μ₂

−β₁f

0, μ₂ 0.

2.10

Hereρx > 0,γ > 0.In particular, the choiceμ₂ μ₁at2.10implies thatμ²₁−μ₁² 0, or μ₁ iλ₁, whereλ₁ ≥ 0. Therefore, zeros of the functionϕλcan lie only on the imaginary axis. Now, let us now prove that functionϕλhas zeros in finite numbers. This is obvious ifϕ0/0, because, under this assumption, the set of zeros cannot have limit points. In the general case, since we can give an estimate for the distance between the neighboring zeros of the functionϕλ,it follows that the number of zeros is finitesee2, page 186.

Let

m⁻²_k ≡ _∞

0

ρxfx, iλk²dx 1

γβ2f0, iλk−β1f0, iλk² 1

2iμkγϕiλk

β2f0, iλk−β1f0, iλk

, k1,2, . . . , n.

2.11

These numbers are called the normalized numbers for the boundary problem1.1–1.3.

The collections{Sλ, −∞< λ <∞;λk;mk k1,2, . . . , n}are called the scattering data of the boundary value problem1.1–1.3. The inverse scattering problem consists in recovering the coefficientqxfrom the scattering data.

3. Fundamental Equation or Modified Marchenko Equation

From1.9,1.10, it is clear that in order to determineqxit is sufficient to knowKx, t. To derive the fundamental equation for the kernelKx, tof the solution1.7, we use equality 2.2, which was obtained inLemma 2.1. Substituting expression1.7for fx, λinto this equality, we get

2iλωx, λ

ϕλ −f₀x, λ S₀λf0x, λ

_∞

μxKx, te^−iλtdt S0λ−Sλf0x, λ

_∞

μxKx, tS0λ−Sλe^iλtdt−S0λ _∞

μxKx, te^−iλtdt.

3.1

Multiplying both sides of relation3.1by1/2πe^iλyand integrating overλfrom−∞to∞, for y > μxat the right-hand side we get

K x, y

1 2π

_∞

−∞S0λ−Sλf0x, λe^iλydλ

_∞

μxKx, t 1

2π _∞

−∞S0λ−Sλe^iλtydλ

dt

− _∞

μxKx, t 1

2π _∞

−∞S₀λe^iλtydλ

dt.

3.2

Now we will compute the integral1/2π_∞

−∞S₀λe^iλtydλ. By elementary transforms we obtain

S₀λ e^−2iλa

1−τ² e^2iλaα

1τe^2iλaα τe^−2iλa e^{−2iλa1−α}

1−τ²∞ k0

−1^kτ^ke^2iλaαkτe^−2iλa,

3.3

whereβ₂0. Thus we have

1 2π

_∞

−∞S0λe^iλtydλ

1−τ^2∞

k0

−1^kτ^kδ

ty−2a1−α 2aαk τδ

ty−2a , 3.4

whereδtis the Dirac delta function.

Forβ2/0, similarly we get 1

2π _∞

−∞S₀λe^iλtydλ

τ²−1∞ k0

−1^kτ^kδ

ty−2a1−α 2aαk τδ

ty−2a . 3.5 Consequently,3.2can be written as

K x, y

FS

x, y

_∞

μxKx, tF0S

ty

dt−τK

x,2a−y

−

1−τ^2∞

k0

−1^kτ^kK

x,2a1−α−2aαk−y ,

3.6

where

F_0Sx≡ 1 2π

_∞

−∞S0λ−Sλe^iλxdλ, FS

x, y

≡ 1

2 1 1

ρx

F0S

μx y

1

2 1− 1 ρx

F0S

μ⁻x y .

3.7

Let us show that for y > μx the last expression in the sum equals zero. We note that Kx, z 0 forz < x. Fory > μxwe have

2a1−α−2aαk−y < μx, k0,1,2, . . . . 3.8

If 0< x < a,thenμx αx−αaa,and hence

2a1−α−2aαk−y <2a−2aαk1−αxαa−a

a−aα−2aαk−αx < a1−α≤μx. 3.9

Ifx≥a, thenμx x,and hence, for this case, the inequality holds.

Therefore, fory > μx 3.2takes the form

K x, y

F_S x, y

_∞

μxKx, tF0S

ty

dt1− ρx

1

ρxK

x,2a−y

. 3.10

On the left-hand side of3.1with help of Jordan’s lemma and the residue theorem and by takingLemma 2.2into account fory > μx,we obtain

−ⁿ

k1

2iλkωx, iλk

ϕiλk e^−λky. 3.11

From the definition of normalized numbersm_kk1,2, . . . , nin2.11we have

−ⁿ

k1

2iλ_kωx, iλke^−λky

ϕiλk −ⁿ

k1

2iλke^−λkyfx, iλk β₂f0, iλk−β₁f0, iλk

ϕiλk −ⁿ

k1

m²_kfx, iλke^−λky

−ⁿ

k1

m²_k

f₀x, iλke^−λkxy _∞

μxKx, te^−λktydt

.

3.12

Thus, fory > μx,by taking3.10and3.12into account, from3.2we derive the relation

−ⁿ

k1

m²_k f₀x, iλke^−λkxy _∞

μxKx, te^−λktydt

!

FS

x, y

_∞

μxKx, tF0S

ty dtK

x, y

1− ρx

1

ρxK

x,2a−y .

3.13

Consequently, we obtain fory > μx

F x, y

_∞

μxKx, tF0

ty dtK

x, y

1− ρx

1

ρxK

x,2a−y

0, 3.14

where

F₀x F_0Sxⁿ

k1

m²_ke^−λkx 1 2π

_∞

−∞S0λ−Sλe^−λkxdλⁿ

k1

m²_ke^−λkx,

F x, y

FS

x, y ⁿ

k1

m²_kf0x, iλke^−λkxy

1

2 1 1

ρx

F₀

yμx 1

2 1− 1 ρx

F₀

yμ⁻x .

3.15

Equation 3.14 is called the fundamental equation of the inverse problem of the scattering theory for the boundary problem1.1–1.3. The fundamental equation is different from the classic equation of Marchenko and we call equation3.14the modified Marchenko equation.

The discontinuity of the functionρxstrongly influences the structure of the fundamental equation of the boundary problem1.1–1.3.

Thus, we have proved the following theorem.

Theorem 3.1. For eachx≥0, the kernelKx, yof the special solution1.7satisfies the fundamental equation3.14.

By using the fundamental equation it is shown that the scattering function Sλ is continuous at all real pointsλand

S0

⎧⎨

⎩

1, ifϕ0/0,

−1, ifϕ0 0. 3.16

It can be shown thatS0λ−Sλtends to zero as|λ| → ∞and is the Fourier transform of some function inL₂−∞,∞.

4. Solvability of the Fundamental Equation

Substituting scattering data into 3.15, we construct F₀x and Fx, y. The fundamental equation3.14can be written in the more convenient form

K

x, tμx qK

x,2a−t−μx F

x, tμx

_∞

0

K

x, ξμx F₀

ξt2μx

dξ0, t >0. 4.1

We will seek the solutionKx, tμxof4.1for everyx≥0 in the same spaceL10,∞.

We consider the operatorsF^S_0x,F0xacting in the spacesL_i0,∞ i1,2, respectively, by the rules

F^S_0xf _∞

0

F_0S

ξt2μx fξdξ,

F0xf _∞

0

F0

ξt2μx fξdξ

4.2

which appear in the fundamental equation.

The operatorsF^S_0x,F0xare compact in each spaceLi0,∞ i1,2for every choice of μx ≥0. The proof of this fact completely repeats the proof of Lemma 3.3.1 which can be found in2.

Substitutingft≡Kx, t2μxinto4.1, we obtain ft τTft F0xft F

x, tμx

0, 4.3

where

Tft f2a−t. 4.4

In order to prove the solvability of the given fundamental equation, it suffices to verify that the homogenous equation

ft τTft F0xft 0 4.5

has no nontrivial solutions in the corresponding space.

From the homogenous equation4.5we obtain τT

ft τTft

τTF0xft 0, 4.6

and, sinceT²I,we have

τTft −τ²ft−τTF0xft. 4.7

Using this equality in4.5, we have ft τTft F0xft

1−τ²

ft I−τTF0xft 0, 4.8

or takingft f,we obtain the equation f− 1

1−τ²I−τTF0xf0 4.9

from which4.5is obtained.

Theorem 4.1. Equation4.5has a unique solutionKx,·∈L₁μx,∞for each fixedx≥0.

To prove this theorem we need some of auxiliary lemmas.

Lemma 4.2. If ft ∈ L10,∞ is a solution of the homogenous equation 4.5, then ft ∈ L_∞0,∞.

Proof. In fact, the kernelF₀ξt2μxofF0xcan be approximated by a bounded function Φξt2μx,so that_∞

0 |F0t−Φt|dt <1. By rewriting4.5in the form f− 1

1−τ²I−τTΦ−F0xf− 1

1−τ²I−τTΦf, 4.10

we obtain an equation with a bounded function on the right-hand side, where

Φf≡Φft _∞

0

fξΦ

ξt2μx

dξ. 4.11

In the spaceL_∞L_∞0,∞we get

""Φ−F0xf""

L∞≤ fL_∞

_∞

0

Φ

ξt2μx

−F₀

ξt2μx dξ f_L∞

_∞

t2μξ|Φξ−F0ξ|dξ≤ fL∞.

4.12

Hence

""

"" 1

1−τ²I−τTΦ−F0x""

""

L∞→L∞

≤ 1τ

1−τ²Φ−F0x_L∞→L∞ ≤1. 4.13 Thus, the function on the right-hand side of 4.10is bounded. Consequently, we havef φ#_∞

n1Bⁿφ, where φ− 1

1−τ²I−τTΦf, B 1

1−τ²I−τTΦ−F0x, 4.14

and the series converges in L10,∞ as well as in L_∞0,∞; that is, the solution of the homogenous equation4.5is bounded.

Corollary 4.3. If ft ∈ L₁0,∞ is a solution of the homogenous equation 4.5, then ft ∈ L20,∞.

Proof. In fact,ft∈L₁0,∞∩L_∞0,∞⊂L₂0,∞.

Thus, it suffices to investigate4.5in the spaceL₂0,∞.

Lemma 4.4. The operators IτTF0xacting inL20,∞are nonnegative for everyμx≥0:

IτTF0xf,f≥0, 4.15

and equality is attained if and only if

f$λ−Sλe^2iλμxf−λ $ 0, −∞< λ <∞,

f−iλ$ k 0, k1,2, . . . , n, 4.16

wherefλ$ is Fourier transform of the functionft.

Proof. According to definitions of the operatorsF0xandT we get

IτTF0xf,f f²τ _∞

0

ftf

2a−t2μx dt 1

2π _∞

−∞S0λ−Sλe^2iλμxf−λ$ fλdλ$ 1

2π _∞

−∞

$fλ²dλ− 1 2π

_∞

−∞S0λe^2iλμxf−λ$ f$λdλ 1

2π _∞

−∞S0λ−Sλe^2iλμxf−λ$ fλdλ$ ⁿ

k1

m²_k $f−iλk² 1

2π _∞

−∞

fλ$ −Sλe^2iλμxf−λ$

f$λdλⁿ

k1

m²_k $f−iλk².

4.17

Since|Sλe^2iλμx|1, _∞

−∞Sλe^2iλμxf−λ$ fλdλ$ ²≤

_∞

−∞

$f−λ²dλ _∞

−∞

$fλ²dλ 4.18

by the Cauchy-Bunyakovskii inequality, or, equivalently, _∞

−∞Sλe^2iλμxf−λ$ fλdλ$ ≤

_∞

−∞

$fλ²dλ. 4.19

Therefore, the first term on the right-hand side of formula4.17is nonnegative. Since the second term is obviously nonnegative. Inequality4.16holds, with equality, if and only if

f−iλ$ k 0 k1,2, . . . , n, _∞

−∞

fλ$ −Sλe^2iλμxf$−λ

fλdλ$ 0.

4.20

This shows that the function zλ fλ$ − Sλe^2iλμxf−λ$ is orthogonal to fλ$ in L₂−∞,∞.But then

""

" $fλ"""²"""Sλe^2iλμxf−λ$ """²""" $fλ−zλ"""²""" $fλ"""²zλ², 4.21 which is possible if and only if zλ 0. Thus, inequality4.15 holds, with equality for those functionsftwhose Fourier transformfλ$ satisfies conditions4.16. The lemma is proved.

With the help of Lemmas4.2and4.4, we obtain the proof ofTheorem 4.1. It remains to show that the homogenous equation4.5has only the null solution inL20,∞.But, by Lemma 4.4the Fourier transformfλ$ of any solutionftof4.5satisfies the identityf$λ− Sλe^2iλμxf$−λ 0.Hence, upon settingϕ$hλ fλe$ ^−iλx√

ρxcosλh, 0< h < x

ρx, we get

$

ϕ_hλ−Sλe^2iλa1−αϕ$_h−λ 0. 4.22

Sinceϕ$hλis the Fourier transform of the function ϕ_ht 1

2

% f

th−x ρx

f

t−h−x ρx

&

, 4.23

which vanishes fort < x

ρx−h,identity4.22yields

ϕ_ht τϕ_h2a−t _∞

0

ϕ_hξFosξtdξ0 4.24

for all h ∈ 0, x

ρx.Therefore, if 4.5has nonzero solution, 4.24has infinitely many linear independent solutionsϕht,which in turn contradicts the compactness of the operator F0x.Hence,ft 0.

According to Theorems3.1and4.1the following result holds.

Theorem 4.5. The scattering data uniquely determine the boundary value problem1.1–1.3.

Proof. To form the fundamental equation 3.14, it suffices to know the functions F₀x and Fx, y. In turn, to find the functions F0x, Fx, y, it suffices to know only the scattering data {Sλ −∞ < λ < ∞;λ_k, m_k k 1,2, . . . , n}. Given the scattering data,

we can use formulas 3.15 to construct the functions F0x, Fx, y and write out the fundamental equation3.14for the unknown functionKx, y. According toTheorem 4.1, the fundamental equation has a unique solution. Solving this equation, we find the kernel Kx, y of the special solution 1.7, and hence, according to formulas 1.9-1.10, it is constructed the potentialqx.

Remark 4.6. In the case when ρxis a positive piecewise-constant with a finite number of points of discontinuity, similar results can be obtained.

Acknowledgment

This research is supported by the Scientific and Technical Research Council of Turkey.

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