The Matrix Schrödinger Operator and Schrödinger Operator on Graphs

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dc.contributor.advisor Pavlov, B. S en
dc.contributor.author Harmer, Mark Stuart en
dc.date.accessioned 2007-07-10T01:35:20Z en
dc.date.available 2007-07-10T01:35:20Z en
dc.date.issued 2000 en
dc.identifier THESIS 00-481 en
dc.identifier.citation Thesis (PhD--Mathematics)--University of Auckland, 2000 en
dc.identifier.uri http://hdl.handle.net/2292/786 en
dc.description Full text is available to authenticated members of The University of Auckland only. en
dc.description.abstract 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 [50] 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 [19] 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. en
dc.language.iso en en
dc.publisher ResearchSpace@Auckland en
dc.relation.ispartof PhD Thesis - University of Auckland en
dc.relation.isreferencedby UoA9993519114002091 en
dc.rights Restricted Item. Available to authenticated members of The University of Auckland. en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.title The Matrix Schrödinger Operator and Schrödinger Operator on Graphs en
dc.type Thesis en
thesis.degree.grantor The University of Auckland en
thesis.degree.level Doctoral en
thesis.degree.name PhD en
dc.rights.holder Copyright: The author en


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