Abstract
The inverse scattering problem for the one-dimensional Schrödinger equation is considered when the potential is real valued and integrable and has a finite first-moment and no bound states. Corresponding to such potentials, for rational reflection coefficients with only simple poles in the upper half complex plane, a method is presented to recover the potential and the scattering solutions explicitly. A numerical implementation of the method is developed. For such rational reflection coefficients, the scattering wave solutions to the plasma-wave equation are constructed explicitly. The discontinuities in these wave solutions and in their spatial derivatives are expressed explicitly in terms of the potential.
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.
References
T. Aktosun and M. Klaus, Chapter 2.2.4, Inverse theory: problem on the line, in: E.R. Pike and P.C. Sabatier (eds.), Scattering, Academic Press, London, 2001, pp.770–785.
K. Chadan and P.C. Sabatier, Inverse problems in quantum scattering theory, 2nd ed., Springer, New York, 1989.
P. Deift and E. Trubowitz, Inverse scattering on the line, Comm. Pure Appl. Math. 32 (1979), 121–251.
L.D. Faddeev, Properties of the S-matrix of the one-dimensional Schrodinger equation, Am. Math. Soc. Transl. (ser. 2) 65 (1967), 139–166.
V.A. Marchenko, Sturm-Liouville operators and applications, Birkhäuser, Basel, 1986.
T. Aktosun, M. Klaus, and C. van der Mee, Explicit Wiener-Hopf factorization for certain nonrational matrix functions, Integral Equations Operator Theory 15 (1992), 879–900.
B. Dolveck-Guilpart, Practical construction of potentials corresponding to exact rational reflection coefficients, in: P.C. Sabatier (ed.), Some topics on inverse problems, World Sci. Publ., Singapore, 1988, pp. 341–368.
I. Kay, The inverse scattering problem when the reflection coefficient is a rational function, Comm. Pure Appl. Math. 13 (1960), 371–393.
K.R. Pechenick and J.M. Cohen, Inverse scattering — exact solution of the Gel’fand-Levitan equation, J. Math. Phys. 22 (1981), 1513–1516.
K.R. Pechenick and J.M. Cohen, Exact solutions to the valley problem in inverse scattering, J. Math. Phys. 24 (1983), 406–409.
R.T. Prosser, On the solutions of the Gel’fand-Levitan equation, J. Math. Phys. 25 (1984), 1924–1929.
P.C. Sabatier, Rational reflection coefficients in one-dimensional inverse scattering and applications, in: J.B. Bednar et al. (eds.), Conference on inverse scattering: theory and application, SIAM, Philadelphia, 1983, pp. 75–99.
P.C. Sabatier, Rational reflection coefficients and inverse scattering on the line, Nuovo Cimento B 78 (1983), 235–248.
P.C. Sabatier, Critical analysis of the mathematical methods used in electromagnetic inverse theories: a quest for new routes in the space of parameters, in: W. M. Boerner et al. (eds.), Inverse methods in electromagnetic imaging, Reidel Publ., Dordrecht, Netherlands, 1985, pp. 43–64.
D. Alpay and I. Gohberg, Inverse problem for Sturm-Liouville operators with rational reflection coefficient, Integral Equations Operator Theory 30 (1998), 317–325.
C. van der Mee, Exact solution of the Marchenko equation relevant to inverse scattering on the line, in: V.M. Adamyan et al. (eds.), Differential operators and related topics, Vol. I, Birkhäuser, Basel, 2000, pp. 239–259.
R. Courant and D. Hilbert, Methods of mathematical physics, Vol. I, Interscience Publ., New York, 1953.
T. Aktosun and J.H. Rose, Wave focusing on the line, J. Math. Phys. 43 (2002), 3717–3745.
P.E. Sacks, Reconstruction of steplike potentials, Wave Motion 18 (1993), 21–30.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Israel Gohberg on the occasion of his 75th birthday
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Aktosun, T., Borkowski, M.H., Cramer, A.J., Pittman, L.C. (2005). Inverse Scattering with Rational Scattering Coefficients and Wave Propagation in Nonhomogeneous Media. In: Gohberg, I., et al. Recent Advances in Operator Theory and its Applications. Operator Theory: Advances and Applications, vol 160. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7398-9_1
Download citation
DOI: https://doi.org/10.1007/3-7643-7398-9_1
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7290-3
Online ISBN: 978-3-7643-7398-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)