Inverse scattering on the line

Deift, Percy; Trubowitz, Eugene

doi:10.1002/cpa.3160320202

Cited by 803 publications

(834 citation statements)

References 14 publications

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“…Faddeev [21] and Delft and Trubowitz [20] for the SchrSdinger equation on the line, we derive the analyticity and asymptotic properties of the Faddeev matrices and the scattering coefficients, employ them to derive a Riemann-Hilbert problem and various Marchenko integral equations, and recover the potential in terms of the solutions of the Marchenko equations. We prove the unitarity of the scattering matrix and exploit this property to prove the unique solvability of the Marchenko equations.…”

Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

“…After that, for rational reflection coefllcients we present a procedure to compute explicitly the scattering matrix from a reflection coefficient. This is no longer as elementary as in the case of the (scalar) SchrSdinger equation [20,21] and involves a suitable extension of a contractive n x n matrix function to a unitary 2n x 2n matrix function (cf. [8,27]).…”

Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

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Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

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A direct and inverse scattering theory on the full line is developed for a class of firstorder selfadjoint 2n • 2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

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Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

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“…-To the best of my knowledge, formulas (3.7) and (3.14) have not appeared in the literature on the Toda lattice. However, related issues have been considered for semi-infinite Jacobi matrices in [8], and for the inverse scattering on the line in [9]. It seems to me interesting that the curve theoretic approach to the orthogonality relations applies when the curves have worse singularities than double points, which leads to nonselfadjoint eigenvalue problems, see [22].…”

Section: Corollary 34 -In the Basis Of The Discrete Jost Functions mentioning

confidence: 99%

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

Haine¹

2005

Annales De L’institut Fourier

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“…We note the following restatement of a theorem of Levinson -see Deift and Trubowitz [16] for extensive discussion and proofs. …”

Section: (Y Rcmentioning

confidence: 99%

Scattering theory for systems with different spatial asymptotics on the left and right

1978

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Abstract. We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4-V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x > 0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, "the twisting trick", is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.

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Inverse scattering on the line

Cited by 803 publications

References 14 publications

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

The spectral matrices of Toda solitons and the fundamental solution of some discrete heat equations

Scattering theory for systems with different spatial asymptotics on the left and right

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