The Matrix Schrödinger Operator and Schrödinger Operator on Graphs
Abstract (Summary)Restricted Item. Print thesis available in the University of Auckland Library or available through Inter-Library Loan. We explicitly parameterise the Lagrange planes of a hermitian symplectic space in terms of U(n) consequently describing the self-adjoint boundary conditions in terms of a unitary matrix. For general self-adjoint boundary conditions the scattering matrix has non-standard asymptotics which are derived. A factorisation of the matrix schrödinger operator and hence a Darboux transformation is found with the property that it transforms the potential as well as the boundary conditions. We describe the inverse scattering problem for the matrix Schrödinger operator with general self-adjoint boundary conditions in terms of a Marchenko equation as well as a Riemann-Hilbert problem. Self-adjoint boundary conditions may be viewed as inducing a zero-range potential at the origin  and we show that from the scattering data we may recover the matrix potential as well as the boundary conditions/zero-range potential at the origin. In the case of diagonal potential—identifiable with the Schrödinger operator on the graph—we need only the n diagonal elements of the scattering data to recover the potential as well as the boundary conditions/zero-range potential at the origin. Given the boundary conditions we may reduce the set of scattering data necessary to recover the potential. In the case of flux-conserved boundary conditions [37, 38] it is shown that we need only n-1 diagonal elements of the scattering data to recover the potential. Using the construction of the asymptotic hermitian symplectic space [65, 66] we derive a formula for the scattering matrix of a graph in terms of the scattering matrices of its subgraphs. This also provides a characterisation of the discrete eigenvalues embedded in the continuous spectrum. For the Laplacean on a compact graph we provide a description of the spectrum in terms of the geometry of the graph. Finally we generalise the asymptotic formula for the scattering matrix in  to the case of non-simple spectrum. we use this asymptotic formula to identify a simple family of switches and investigate the properties of a member of the family using numerical techniques.
School Location:New Zealand
Source Type:Master's Thesis
Date of Publication:11/07/2000