Abstract
We study the n-dimensional Schrödinger equation, n ≥ 2, with a nonspherically symmetric potential in the class of Agmon’s short range potentials without any positive energy bound states. We give sufficient conditions that guarantee the existence of a Wiener-Hopf factorization of the corresponding scattering operator. We show that the potential can be recovered from the scattering operator by solving a related Riemann-Hilbert problem utilizing the Wiener-Hopf factors of the scattering operator. We also study the properties of the scattering operator and show that it is a trace class perturbation of the identity when the potential is also integrable.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Literature
S. Agmon, Spectral Properties of Schrödinger Operators and Scattering Theory, Ann. Scuola Norm. Sup. Pisa 2, 151–218 (1975).
T. Aktosun and C. van der Mee, Solution of the Inverse Scattering Problem for the 3-D Schrödinger Equation by Wiener-Hopf Factorization of the Scattering Operator, J. Math. Phys., to appear.
T. Aktosun and C. van der Mee, Solution of the Inverse Scattering Problem for the 3-D Schrödinger Equation using a Fredholm Integral Equation, Preprint.
H. Bart, I. Gohberg and M. A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Birkhäuser OT 1, Basel and Boston, 1979.
H. Bart, I. Gohberg and M. A. Kaashoek, Explicit Wiener-Hopf Factorization and Realization. In: I. Gohberg and M. A. Kaashoek, Constructive Methods of Wiener-Hopf Factorization, Birkhäuser OT 21, Basel and Boston, 1986, pp. 235-316.
R. Beals and R. R. Coifman, Multidimensional Inverse Scattering and Nonlinear P.D.E.’s, Proc. Symp. Pure Math. 43, 45–70 (1985).
R. Beals and R. R. Coifman, The D-bar Approach to Inverse Scattering and Nonlinear Evolutions, Physica D 18, 242–249 (1986).
K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, Second Edition, Springer, New York, 1989.
L.D. Faddeev, Increasing Solutions of the Schrödinger Equation, Sov. Phys. Dokl. 10, 1033–1035 (1965) [Dokl. Akad. Nauk SSSR 165, 514-517 (1965) (Russian)].
L.D. Faddeev, Three-dimensional Inverse Problem in the Quantum Theory of Scattering, J. Sov. Math. 5, 334–396 (1976) [Itogi Nauki i Tekhniki 3, 93-180 (1974) (Russian)].
I.C. Gohberg, The Factorization Problem for Operator Functions, Amer. Math. Soc. Transl., Series 2, 49, 130–161 (1966) [Izvestiya Akad. Nauk SSSR, Ser. Matem., 28, 1055-1082 (1964) (Russian)].
I.C. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monographs, Vol. 18, A.M.S., Providence, 1969 [Nauka, Moscow, 1965 (Russian)].
I.C. Gohberg and J. Leiterer, Factorization of Operator Functions with respect to a Contour. III. Factorization in Algebras, Math. Nachrichten 55, 33–61 (1973) (Russian).
T. Kato, Growth Properties of Solutions of the Reduced Wave Equation with a Variable Coefficient, Comm. Pure Appl. Math. 12, 403–425 (1959).
S. Kuroda, An Introduction to Scattering Theory, Lecture Notes Series, Vol. 51, Math. Inst., Univ. of Aarhus, 1980.
N.I. Muskhelishvili, Singular Integral Equations, Noordhoff, Groningen, 1953 [Nauka, Moscow, 1946 (Russian)].
A.I. Nachman and M.J. Ablowitz, A Multidimensional Inverse Scattering Method, Studies in Appl. Math. 71, 243–250 (1984).
R.G. Newton, The Gel’fand-Levitan Method in the Inverse Scattering Problem in Quantum Mechanics. In: J.A. Lavita and J.-P. Marchand (Eds.), Scattering Theory in Mathematical Physics, Reidel, Dordrecht, 1974, pp. 193–225.
R.G. Newton, Inverse Scattering. II. Three Dimensions, J. Math. Phys. 21, 1698–1715 (1980); 22, 631 (1981); 23, 693 (1982).
R.G. Newton, Inverse Scattering. III. Three Dimensions, Continued, J. Math. Phys. 22, 2191–2200 (1981); 23, 693 (1982).
R.G. Newton, Inverse Scattering. IV. Three Dimensions: Generalized Marchenko Construction with Bound States, J. Math. Phys. 23, 2257–2265 (1982).
R.G. Newton, A Faddeev-Marchenko Method for Inverse Scattering in Three Dimensions, Inverse Problems 1, 371–380 (1985).
R.G. Newton, Eigenvalues of the S-matrix, Phys. Rev. Lett. 62, 1811–1812 (1989).
R.G. Newton, Inverse Schrödinger Scattering in Three Dimensions, Springer, New York, 1989.
R.G. Novikov and G.M. Henkin, Solution of a Multidimensional Inverse Scattering Problem on the Basis of Generalized Dispersion Relations, Sov. Math. Dokl. 35, 153–157 (1987) [Dokl. Akad. Nauk SSSR 292, 814-818 (1987) (Russian)].
R.T. Prosser, Formal Solution of Inverse Scattering Problems, J. Math. Phys. 10, 1819–1822 (1969).
R.T. Prosser, Formal Solution of Inverse Scattering Problems. II, J. Math. Phys. 17, 1775–1779 (1976).
R.T. Prosser, Formal Solution of Inverse Scattering Problems. III, J. Math. Phys. 21, 2648–2653 (1980).
R.T. Prosser, Formal Solution of Inverse Scattering Problems. IV, J. Math. Phys. 23, 2127–2130 (1982).
M. Schechter, Spectra of Partial Differential Operators, North-Holland, Amsterdam, 1971.
R. Weder, Multidimensional Inverse Scattering Theory, Inverse Problems, in press (April 1990).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Basel AG
About this chapter
Cite this chapter
Aktosun, T., van der Mee, C. (1991). Wiener-Hopf Factorization in the Inverse Scattering Theory for the n-D Schrödinger Equation. In: Bart, H., Gohberg, I., Kaashoek, M.A. (eds) Topics in Matrix and Operator Theory. Operator Theory: Advances and Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5672-0_1
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5672-0_1
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5674-4
Online ISBN: 978-3-0348-5672-0
eBook Packages: Springer Book Archive