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On uniqueness in some physical systems

Symons, Frederick 2017. On uniqueness in some physical systems. PhD Thesis, Cardiff University.
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In this work we present some uniqueness and cloaking results for a related pair of inverse problems. The first concerns recovering the parameter q in a Bessel-type operator pencil, over L^2(0, 1; rdr) from (a generalisation of) the Weyl–-Titchmarsh boundary m-function. We assume that both coefficients, w and q, are singular at 0. We prove q is uniquely determined by the sequence m(-n^2) (n = 1, 2, 3, ...), using asymptotic and spectral analysis and m-function interpolation results. For corollary we find, in a halfdisc with a singular “Dirichlet-point” boundary condition on the straight edge, a singular radial Schroedinger potential is uniquely determined by Dirichlet-to- Neumann boundary measurements on the semi-circular edge. The second result concerns recovery of three things—a Schroedinger potential in a planar domain, a Dirichlet-point boundary condition on part of the boundary, and a self-adjointness-imposing condition—from Dirichlet-to-Neumann measurements on the remaining boundary. With modern approaches to the inverse conductivity problem and a solution-space density argument we show the boundary condition cloaks the potential and vice versa. Appealing to negative eigen-value asymptotics we find the full-frequency problem has full uniqueness.

Item Type: Thesis (PhD)
Date Type: Completion
Status: Unpublished
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Funders: EPSRC
Date of First Compliant Deposit: 18 August 2017
Last Modified: 15 Sep 2021 14:59

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