Sonja Currie
University of the Witwatersrand
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Featured researches published by Sonja Currie.
Inverse Problems | 2007
Sonja Currie; Bruce A. Watson
We consider inverse nodal problems on graphs. Eigenfunction and eigenvalue asymptotic approximations are used to provide an asymptotic expression for the spacing of nodal points on each edge of the graph. Based on this, the uniqueness of the potential for given nodal data is proved and we give a construction of q as a limit, in , of a sequence of functions whose nth term is dependent only on the nth eigenvalue and its associated nodal data.
Proceedings of the Royal Society of Edinburgh: Section A Mathematics | 2009
Sonja Currie; Bruce A. Watson
AbstractWe consider an inverse spectral problemfor Sturm-Liouville boundaryvalue problemson a graph with formally self-adjoint boundary conditions at the nodes, where the giveninformation is the M-matrix. Based on the results found in S. Currie, B.A. Watson,M-matrix asymptotics for Sturm-Liouville problems on graphs, J. Com. Appl. Math.,doi: 10.1016/j.cam.2007.11.019, using the Green’s function, we prove that the poles ofthe M-matrix are at the eigenvalues of the associated boundary value problem and aresimple, located on the real axis and that the residue at a pole is a negative semi-definitematrix with rank equal to the multiplicity of the eigenvalue. We define the so callednorming constants and relate them to the spectral measure and the M-matrix. Thisenables us to recover, from the M-matrix, the boundary conditions and the potential,up to a unitary equivalence for co-normal boundary conditions. ∗ Keywords: inverse problem, differential operators on graphs, m-function, Sturm-Liouville (2000)MSC:34A55, 34B45, 34B20, 34L05, 34B27 .
Journal of Difference Equations and Applications | 2013
Sonja Currie; Anne D. Love
This paper continues the work done in [7] in which, for eigenparameter-dependent boundary conditions, it was shown how, by making use of a Crum-type transformation, it is possible to go up and down a hierarchy of difference boundary value problems. In particular, in [7], the terminal end boundary condition is assumed to be of the formwhere is the eigenparameter and is a negative Nevanlinna function with g < 0 and for . In this work the case of g = 0 and consequently of h = 0 or is considered. It is shown how many eigenvalues such a boundary value problem will have and in addition the resulting transformed boundary value problem together with its spectrum is given.
Operators and Matrices | 2011
Sonja Currie; Bruce A. Watson
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This leads to a max-min principle and as a consequence we can formulate an analogue of Dirichlet-Neumann bracketing and this in turn gives rise to asymptotic approximations for the eigenvalues.
Advances in Difference Equations | 2010
Sonja Currie; Anne D. Love
We consider a general weighted second-order difference equation. Two transformations are studied which transform the given equation into another weighted second order difference equation of the same type, these are based on the Crum transformation. We also show how Dirichlet and non-Dirichlet boundary conditions transform as well as how the spectra and norming constants are affected.
Mathematical Methods in The Applied Sciences | 2015
Sonja Currie; Thomas Tobias Roth; Bruce A. Watson
First-order systems in on with absolutely continuous real symmetric π-periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet-type boundary value problems on [0,π] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet-type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright
Boundary Value Problems | 2011
Sonja Currie; Anne D. Love
This paper generalises the work done in Currie and Love (2010), where we studied the effect of applying two Crum-type transformations to a weighted second-order difference equation with various combinations of Dirichlet, non-Dirichlet, and affine -dependent boundary conditions at the end points, where is the eigenparameter. We now consider general -dependent boundary conditions. In particular we show, using one of the Crum-type transformations, that it is possible to go up and down a hierarchy of boundary value problems keeping the form of the second-order difference equation constant but possibly increasing or decreasing the dependence on of the boundary conditions at each step. In addition, we show that the transformed boundary value problem either gains or loses an eigenvalue, or the number of eigenvalues remains the same as we step up or down the hierarchy.
Quaestiones Mathematicae | 2017
Sonja Currie; Anne D. Love
Abstract This paper inductively investigates an inverse problem for difference boundary value problems with boundary conditions that depend quadratically on the eigenparameter. In particular, given the eigenvalues and the weights, we provide an algorithm to uniquely reconstruct the potential.
Mediterranean Journal of Mathematics | 2015
Sonja Currie; Marlena Nowaczyk; Bruce A. Watson
The inverse scattering problem for Sturm–Liouville operators on the line with a matrix transfer condition at the origin is considered. We show that the transfer matrix can be reconstructed from the eigenvalues and reflection coefficient. In addition, for potentials with compact essential support, we show that the potential can be uniquely reconstructed.
Quaestiones Mathematicae | 2014
Sonja Currie; Anne D. Love
Abstract This paper provides an illustration of the work done in [14] where a hierarchy of difference boundary value problems was developed. In particular, we studied the effect of applying a Crum-type transformation to a weighted second order difference equation with general λ-dependent boundary conditions at the end points, for eigenparameter λ. In this paper we demonstrate by means of examples, how it is possible to go up and down the hierarchy of boundary value problems developed in [14], keeping the form of the second order difference equation constant but possibly increasing or decreasing the dependence on λ of the boundary conditions at each step. In addition, the transformed boundary value problem either gains or loses an eigenvalue or the number of eigenvalues remains the same as we step up or down the hierarchy.