Abstract
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 self-adjoint 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 self-adjoint 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 bound-state 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 one-to-one 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Z.S. Agranovich, V.A. Marchenko, The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963)
T. Aktosun, M. Klaus, R. Weder, Small-energy analysis for the self-adjoint matrix Schrödinger operator on the half line. J. Math. Phys. 52, 102101 (2011)
T. Aktosun, R. Weder, High-energy analysis and Levinson’s theorem for the selfadjoint matrix Schrödinger operator on the half line. J. Math. Phys. 54, 012108 (2013)
P. Deift, E. Trubowitz, Inverse scattering on the line. Commun. Pure Appl. Math. 32, 121–251 (1979)
L.D. Faddeev, Properties of the S-matrix of the one-dimensional Schrödinger equation. Trudy Mat. Inst. Steklov 73, 314–336 (1964, in Russian) [Am. Math. Soc. Transl. (Ser. 2) 65, 139–166 (1967) (English translation)]
M.S. Harmer, Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions. ANZIAM J. 44, 161–168 (2002)
M.S. Harmer, The matrix Schrödinger operator and Schrödinger operator on graphs. Ph.D. Thesis. University of Auckland, Auckland (2004)
M.S. Harmer, Inverse scattering on matrices with boundary conditions. J. Phys. A 38, 4875–4885 (2005)
V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires. J. Phys. A 32, 595–630 (1999)
V. Kostrykin, R. Schrader, Kirchhoff’s rule for quantum wires. II: The inverse problem with possible applications to quantum computers. Fortschr. Phys. 48, 703–716 (2000)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Aktosun, T., Weder, R. (2021). The Matrix Schrödinger Equation and the Characterization of the Scattering Data. In: Direct and Inverse Scattering for the Matrix Schrödinger Equation. Applied Mathematical Sciences, vol 203. Springer, Cham. https://doi.org/10.1007/978-3-030-38431-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-38431-9_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38430-2
Online ISBN: 978-3-030-38431-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)