On an Inverse Scattering Problem for a Discontinuous Sturm-Liouville Equation with a Spectral Parameter in the Boundary Condition
© Khanlar R. Mamedov. 2010
Received: 9 April 2010
Accepted: 22 May 2010
Published: 21 June 2010
An inverse scattering problem is considered for a discontinuous Sturm-Liouville equation on the half-line 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.
is a positive piecewise-constant function with a finite number of points of discontinuity, are real numbers, and
The aim of the present paper is to investigate the direct and inverse scattering problem on the half-line for the boundary value problem (1.1)–(1.3). In the case , the inverse problem of scattering theory for (1.1) with boundary condition not containing spectral parameter was completely solved by Marchenko [1, 2], Levitan [3, 4], Aktosun , as well as Aktosun and Weder . The discontinuous version was studied by Gasymov  and Darwish . In these papers, solution of inverse scattering problem on the half-line by using the transformation operator was reduced to solution of two inverse problems on the intervals and . In the case , the inverse scattering problem was solved by Guseĭnov and Pashaev  by using the new (nontriangular) representation of Jost solution of (1.1). It turns out that in this case the discontinuity of the function strongly 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 in . Uniqueness of the solution of the inverse problem and geophysical application of this problem for (1.1) when were given by Tihonov  and Alimov . Inverse problem for a wave equation with a piecewise-constant coefficient was solved by Lavrent'ev . Direct problem of scattering theory for the boundary value problem (1.1)–(1.3) in the special case was studied in .
When in (1.1) with the spectral parameter appearing in the boundary conditions, the inverse problem on the half-line was considered by Pocheykina-Fedotova  according to spectral function, by Yurko [16–18] according to Weyl function, and according to scattering data in [19, 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  and wave equation by Yurko [16, 17]. Spectral analysis of the problem on the half-line was studied by Fulton .
Also, physical application of the problem with the linear spectral parameter appearing in the boundary conditions on the finite interval was given by Fulton . 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 Freiling , Chugunova , Rundell and Sacks , Guliyev , and other works cited therein.
This paper is organized as follows. In Section 2, the scattering data for the boundary value problem (1.1)–(1.3) are defined. In Section 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 in Section 4.
is the Jost solution of (1.1) when where .
According to Lemma 2.2 in Section 2, the equation has only a finite number of simple roots in the half-plane ; all these roots lie in the imaginary axis. The behavior of this boundary value problem (1.1)–(1.3) is expressed as a self-adjoint eigenvalue problem.
the scattering function for the boundary value problem (1.1)–(1.3), where denotes the complex conjugate of .
where . It turns out that the potential in the boundary value problem (1.1)–(1.3) is uniquely determined by specifying the set of values The set of values is called the scattering data of the boundary value problem (1.1)–(1.3). The inverse scattering problem for boundary value problem (1.1)–(1.3) consists in recovering the coefficient from the scattering data.
for the unknown function . 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 strongly influences the structure of the fundamental equation of the boundary problem (1.1)–(1.3). By Theorem 4.1 in Section 4, the integral equation has a unique solution for every . Solving this equation, we find the kernel of the special solution (1.7), and hence according to formula (1.10) it is constructed the potential
while and are solutions of (1.1) when , satisfying the initial conditions and
The solvability of these integral equations is obtained through the method of successive approximations. By using integral equations (1.22)–(1.24) for equalities (1.9), (1.10) are obtained. By substituting the expressions for the functions and in (1.1), it can be shown that (1.11) holds.
2. The Scattering Data
For real the functions and form a fundamental system of solutions of (1.1) and their Wronskian is computed as . Here the Wronskian is defined as
The following assertion is valid.
The function is called the scattering function of the boundary value problem (1.1)–(1.3).
The function may have only a finite number of zeros in the half-plane . Moreover, all these zeros are simple and lie in the imaginary axis.
Since for all real , the point is the possible real zero of the function . Using the analyticity of the function in upper half-plane and the properties of solution (1.7) are obtained that zeros of form at most countable and bounded set having as the only possible limit point.
Here , In particular, the choice at (2.10) implies that , or , where . Therefore, zeros of the function can lie only on the imaginary axis. Now, let us now prove that function has zeros in finite numbers. This is obvious if , 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 function it follows that the number of zeros is finite (see [2, page 186]).
These numbers are called the normalized numbers for the boundary problem (1.1)–(1.3).
The collections are called the scattering data of the boundary value problem (1.1)–(1.3). The inverse scattering problem consists in recovering the coefficient from the scattering data.
3. Fundamental Equation or Modified Marchenko Equation
where is the Dirac delta function.
If , then and hence, for this case, the inequality holds.
Equation (3.14) is called the fundamental equation of the inverse problem of the scattering theory for the boundary problem (1.1)–(1.3). The fundamental equation is different from the classic equation of Marchenko and we call equation (3.14) the modified Marchenko equation. The discontinuity of the function strongly influences the structure of the fundamental equation of the boundary problem (1.1)–(1.3).
Thus, we have proved the following theorem.
For each , the kernel of the special solution (1.7) satisfies the fundamental equation (3.14).
It can be shown that tends to zero as and is the Fourier transform of some function in .
4. Solvability of the Fundamental Equation
We will seek the solution of (4.1) for every in the same space .
which appear in the fundamental equation.
The operators are compact in each space for every choice of . The proof of this fact completely repeats the proof of Lemma which can be found in .
has no nontrivial solutions in the corresponding space.
from which (4.5) is obtained.
Equation (4.5) has a unique solution for each fixed .
To prove this theorem we need some of auxiliary lemmas.
If is a solution of the homogenous equation (4.5), then .
and the series converges in as well as in ; that is, the solution of the homogenous equation (4.5) is bounded.
If is a solution of the homogenous equation (4.5), then .
In fact, .
Thus, it suffices to investigate (4.5) in the space .
where is Fourier transform of the function .
which is possible if and only if . Thus, inequality (4.15) holds, with equality for those functions whose Fourier transform satisfies conditions (4.16). The lemma is proved.
for all Therefore, if (4.5) has nonzero solution, (4.24) has infinitely many linear independent solutions which in turn contradicts the compactness of the operator Hence,
According to Theorems 3.1 and 4.1 the following result holds.
The scattering data uniquely determine the boundary value problem (1.1)–(1.3).
To form the fundamental equation (3.14), it suffices to know the functions and In turn, to find the functions it suffices to know only the scattering data . Given the scattering data, we can use formulas (3.15) to construct the functions and write out the fundamental equation (3.14) for the unknown function . According to Theorem 4.1, the fundamental equation has a unique solution. Solving this equation, we find the kernel of the special solution (1.7), and hence, according to formulas (1.9)-(1.10), it is constructed the potential .
In the case when is a positive piecewise-constant with a finite number of points of discontinuity, similar results can be obtained.
This research is supported by the Scientific and Technical Research Council of Turkey.
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