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By
Aktosun, Tuncay; Weder, Ricardo
In this chapter we describe the basic ingredients of the direct and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition. We show how the analysis of star graphs and the Schrödinger scattering problem on the full line can be reduced to the study of the matrix Schrödinger equation on the half line with some appropriate selfadjoint boundary conditions. To analyze the direct and inverse problems on the half line, we introduce the input data set consisting of a potential and two constant boundary matrices describing the boundary condition. We define the Faddeev class of input data sets by imposing some appropriate restrictions on the input data sets. We introduce the scattering data set consisting of a scattering matrix and the boundstate information. We define the Marchenko class of scattering data sets by imposing some appropriate restrictions on the scattering data sets. The unique solutions to the direct and inverse scattering problems are achieved by establishing a onetoone correspondence between the Faddeev class of input data sets and the Marchenko class of scattering data sets. Various equivalent descriptions of the Marchenko class are introduced. Such equivalent descriptions allow us to present various different but equivalent results for the characterization of the scattering data in the solution to the inverse problem. For the reader’s convenience various equivalent characterization theorems are stated but their proofs are postponed until Chap.
5
.
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By
Aktosun, Tuncay; Weder, Ricardo
In this chapter we analyze the inverse scattering problem of recovery of the corresponding input data set D in the Faddeev class from a scattering data set S in the Marchenko class. We discuss the nonuniqueness arising in the inverse scattering problem if the scattering matrix is defined one way with the Dirichlet boundary condition and in a different way with a nonDirichlet boundary condition, as usually done in the standard literature. We present the solution to the inverse scattering problem by analyzing the solvability of the Marchenko integral equation and the derivative Marchenko integral equation. We give a proof of the main characterization result presented in Theorem
2.6.1
. By establishing the equivalents among various characterization properties, we provide the proofs for various alternate characterization results presented in Sect.
2.7
. We consider the inverse scattering problem from a given scattering matrix without having the boundstate information. From the given scattering matrix alone we show how to construct a scattering data set belonging to the Marchenko class so that the constructed scattering data set can be used as input into a properly posed inverse scattering problem. We present a proof of the two equivalent characterization results stated in Theorems
2.7.9
and
2.7.10
, where the characterization involves the use of Levinson’s theorem. Next, we introduce and present a proof of Parseval’s equality expressing the completeness of the set consisting of the physical solution and the boundstate solutions in the study of the matrix Schrödinger operator on the half line with the general selfadjoint boundary condition. We present the generalized Fourier map and establish its properties. We then prove the characterization result stated in Theorem
2.8.1
involving the use of the generalized Fourier map. Moreover, we consider the characterization of the scattering data when the potentials are restricted to the class
$$L^1_p(\mathbf R^+)$$
for p > 1 instead of only p = 1 in the Faddeev class. Finally, we formulate the characterization of the scattering data in the special case of the purely Dirichlet boundary condition, which allows us to make a comparison and contrast with the characterization result of Agranovich and Marchenko.
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By
Aktosun, Tuncay; Weder, Ricardo
In this introductory chapter the goals of the monograph are described, the contents of the remaining chapters and Appendix
A
are outlined, and the relevant general references in the literature are mentioned. The direct and inverse scattering problems for the matrix Schrödinger equation on the half line with the general selfadjoint boundary condition are viewed as two mappings, and it is indicated how these two mappings become inverses of each other by specifying their domains appropriately. The traditional definition of the scattering matrix in one way with the Dirichlet boundary condition and in a different way with a nonDirichlet boundary condition is criticized, and it is indicated that such a practice makes it impossible to formulate a wellposed inverse scattering problem unless the boundary condition is already known and the Dirichlet and nonDirichlet boundary conditions are not mixed. It is emphasized that the recovery of the boundary condition should be a part of the solution to the inverse scattering problem rather than a part of the inverse scattering data.
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By
Aktosun, Tuncay; Weder, Ricardo
In this chapter the scattering process is described physically and mathematically, and the definition of the scattering operator is provided in terms of the wave operators introduced by Møller. The role of the limiting absorption principle is indicated, and it is shown how the Hamiltonian and its resolvent are related to the potential and the two boundary matrices describing the general selfadjoint boundary condition. The generalized Fourier maps associated with the absolutely continuous spectrum are introduced and their basic properties are outlined. It is shown how the wave operators are related to the generalized Fourier maps and their adjoints and hence how the scattering operator is related to the generalized Fourier maps. It is shown that the scattering matrix defined in terms of the Jost matrix coincides with the scattering matrix derived from the scattering operator. Various other topics are considered such as the properties of the spectral shift function, trace formulas of Buslaev–Faddeev type, and a Bargmann–Birman–Schwinger bound on the number of bound states.
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By
Aktosun, Tuncay; Weder, Ricardo
In this chapter we present the solution to the direct scattering problem for the halfline matrix Schrödinger equation using an input data set consisting of a matrix potential and a selfadjoint boundary condition. We establish the properties of some of the main quantities arising in the direct scattering problem such as the Jost solution, the Jost matrix, the scattering matrix, the physical solution, and the regular solution. We consider various equivalent formulations of the general selfadjoint boundary condition. We construct the selfadjoint realizations of the corresponding matrix Schrödinger operator. We provide a detailed analysis of the lowenergy and highenergy behavior of the Jost and scattering matrices. We show how the general selfadjoint boundary condition is related to the highenergy behavior of the scattering matrix and how the two boundary matrices used in the description of the boundary condition are explicitly constructed from the scattering data set. We analyze the bound states and prove Levinson’s theorem, showing how a change in the phase of the determinant of the scattering matrix is related to the total number of bound states including the multiplicities. We also show how the scattering matrix and the boundstate information are related to the Marchenko integral equation and the derivative Marchenko integral equation.
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By
Aktosun, Tuncay; Weder, Ricardo
In this chapter we illustrate the theory presented earlier via explicitly solved examples. It is shown how the Marchenko integral equation can yield explicitly solved examples when its kernel contains a matrix exponential and hence becomes separable. The necessity of the integrability of the potential is demonstrated when a general selfadjoint boundary condition is used rather than only the Dirichlet boundary condition. The characterization of the scattering data is illustrated by various examples where all the characterization conditions are satisfied or one or more of the conditions are not satisfied. The examples where only one characterization condition fails indicate the independence of the characterization conditions applied. Some examples are provided to illustrate how a solution to the zeroenergy Schrödinger equation is affected by various restrictions on the scattering data. The solution to the inverse scattering problem is illustrated with various explicit examples, and it is demonstrated how the potential, boundary condition, and other relevant quantities are constructed from a given scattering data set. The use of Levinson’s theorem and the generalized Fourier map is also illustrated through some explicit examples. We also demostrate that the Faddeev class of input data sets is optimal for the Marchenko class of scattering data sets. This is done by considering an extended Faddeev class of input data sets in which the potentials decay too slowly at infinity. Some examples provided illustrate that a scattering data set from the Marchenko class may correspond to an infinite number of input data sets in the extended Faddeev class.
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