Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-28T12:52:08.679Z Has data issue: false hasContentIssue false
  • English
  • Français

Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions

Published online by Cambridge University Press:  17 February 2009

M. S. Harmer
M. S. Harmer
Affiliation:
Department of Mathematics, University of Auckland, New Zealand; e-mail: harmer@math.auckland.ac.nz.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrödinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 161 - 168
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Agranovich, Z. S. and Marchenko, V. A., The Inverse Problem of Scattering Theory (Gordon and Breach, New York, 1963).Google Scholar
[2]Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space (Frederick Ungar Publishing, New York, 1966).Google Scholar
[3]Harmer, M. S., “The matrix Schrödinger operator and Schrödinger operator on graphs’, Ph. D. Thesis, University of Auckland, 2000.Google Scholar
[4]Kostrykin, V. and Schrader, R., “Kirchhoff's rule for quantum wires”, J. Phys. A 32(1999) 595–630.CrossRefGoogle Scholar
[5]Novikov, S. P., Manakov, S. V., Pitaevskii, L. P. and Zakharov, V. E., Theory of Solitons: The Inverse Scattering Method (Consultants Bureau, New York, 1984).Google Scholar