Abstract
Explicit Wiener-Hopf factorizations are obtained for a certain class of nonrational 2×2 matrix functions that are related to the scattering matrices for the 1-D Schrödinger equation. The diagonal elements coincide and are meromorphic and nonzero in the upper-half complex plane and either they vanish linearly at the origin or they do not vanish. The most conspicuous nonrationality consists of imaginary exponential factors in the off-diagonal elements.
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Literature
T. Aktosun,Perturbations and Stability of the Marchenko Inversion Method, Ph.D. thesis, Indiana University, Bloomington, 1986.
T. Aktosun,Marchenko Inversion for Perturbations, I., Inverse Problems3, 523–553 (1987).
T. Aktosun,Potentials which Cause the Same Scattering at all Energies in One Dimension, Phys. Rev. Lett.58, 2159–2161 (1987).
T. Aktosun, Examples of Non-uniqueness in One-dimensional Inverse Scattering for which T(k)=O(k3) and O(k4) as k→0, Inverse Problems3, L1-L3 (1987).
T. Aktosun,Exact Solutions of the Schrödinger Equation and the Non-uniqueness of Inverse Scattering on the Line, Inverse Problems4, 347–352 (1988).
T. Aktosun, M. Klaus and C. van der Mee,Scattering and Inverse Scattering in One-dimensional Nonhomogeneous Media, J. Math. Phys.33, 1717–1744 (1992).
T. Aktosun and R. G. Newton,Nonuniqueness in the One-dimensional Inverse Scattering Problem, Inverse Problems1, 291–300 (1985).
T. Aktosun and C. van der Mee,Scattering and Inverse Scattering for the 1-D Schrödinger Equation with Energy-dependent Potentials, J. Math. Phys.32, 2786–2801 (1991).
K. Clancey and I. Gohberg,Factorization of Matrix Functions and Singular Integral Operators, Birkhäuser OT3, 1981.
A. Degasperis and P. C. Sabatier,Extension of the One-dimensional Scattering Theory, and Ambiguities, Inverse Problems3, 73–109 (1987).
P. Deift and E. Trubowitz,Inverse Scattering on the Line, Comm. Pure Appl. Math.32, 121–251 (1979).
L. D. Faddeev,Properties of the S-matrix of the One-dimensional Schrödinger Equation, Amer. Math. Soc. Transl.2, 139–166 (1964); also: Trudy Matem. Inst. Steklova73, 314–336 (1964) [Russian].
I. M. Gel'fand and B. M. Levitan,On the Determination of a Differential Equation from its Spectral Function, Amer. Math. Soc. Transl., Series2, 1, 253–304 (1955); also: Izv. Akad. Nauk SSSR15 (4), 309–360 (1951) [Russian].
I. C. Gohberg and M. G. Krein,Systems of Integral Equations on a Half-line with Kernels depending on the Difference of Arguments, Amer. Math. Soc. Transl., Series2, 14, 217–287 (1960); also: Uspekhi Matem. Nauk SSSR13(2), 3–72 (1959) [Russian].
I. C. Gohberg and J. Leiterer,Factorization of Operator Functions with respect to a Contour. II. Canonical Factorization of Operator Functions Close to the Identity, Math. Nachr.54, 41–74 (1972) [Russian].
A. B. Lebre,Factorization in the Wiener Algebra of a Class of 2×2 Matrix Functions, Integral Equations and Operator Theory12, 408–423 (1989).
A. B. Lebre and A. F. dos Santos,Generalized Factorization for a Class of Nonrational 2×2 Matrix Functions, Integral Equations and Operator Theory13, 671–700 (1990).
V. A. Marchenko,Sturm-Liouville Operators and Applications, Birkhäuser OT22, Basel-Boston-Stuttgart, 1986.
E. Meister and F.-O. Speck,Wiener-Hopf Factorization of Certain Non-rational Matrix Functions in Mathematical Physics. In: H. Dym et al. (Eds.),The Gohberg Anniversary Collection, II., Birkhäuser OT41, Basel-Boston-Stuttgart, 1989, pp. 385–394.
R. G. Newton,Inverse Scattering. I. One Dimension, J. Math. Phys.21, 493–505 (1980).
R. G. Newton,The Marchenko and Gel'fand-Levitan Methods in the Inverse Scattering Problem in One and Three Dimensions. In: J. B. Bednar et al. (Eds.),Conference on Inverse Scattering: Theory and Application, SIAM, Philadelphia, 1983, pp. 1–74.
R. G. Newton,Remarks on Inverse Scattering in One Dimension, J. Math. Phys.25, 2991–2994 (1984).
R. G. Newton,Factorizations of the S-matrix, J. Math. Phys.31, 2414–2424 (1990).
F. S. Teixeira,Generalized Factorization for a Class of Symbols in [PC(R)]2×2, Appl. Anal.36, 95–117 (1990).