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Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators

Brown, Brian Malcolm, Marletta, Marco and Symons, Frederick 2016. Uniqueness from discrete data in an inverse spectral problem for a pencil of ordinary differential operators. Journal of the London Mathematical Society 94 (3) , pp. 793-813. 10.1112/jlms/jdw059

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Abstract

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order. The uniqueness is achieved from a discrete set of data, namely, the values at the points −n2 (n ∈ N) of (a physically appropriate generalization of) the Weyl– Titchmarsh m-function m(λ) for the problem. As a corollary, we establish a uniqueness result for a physically motivated inverse problem inspired by Berry and Dennis (‘Boundary-conditionvarying circle billiards and gratings: the Dirichlet singularity’, J. Phys. A: Math. Theor. 41 (2008) 135203). To achieve these results, we prove a limit-circle analogue to the limit-point m-function interpolation result of Rybkin and Tuan (‘A new interpolation formula for the Titchmarsh– Weyl m-function’, Proc. Amer. Math. Soc. 137 (2009) 4177–4185); however, our proof, using a Mittag-Leffler series representation of m(λ), involves a rather different method from theirs, circumventing the A-amplitude representation of Simon (‘A new approach to inverse spectral theory, I. Fundamental formalism’, Ann. Math. (2) 150 (1999) 1029–1057). Uniqueness of the potential then follows by appeal to a Borg–Marˇcenko argument.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Computer Science & Informatics
Mathematics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Additional Information: Post print pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/0024-6107/ (accessed 27/06/2016) Copyright London Mathematical Society
Publisher: London Mathematical Society
ISSN: 0024-6107
Funders: Engineering and Physical Sciences Research Council
Date of First Compliant Deposit: 23 June 2016
Date of Acceptance: 13 June 2016
Last Modified: 25 Mar 2019 01:48
URI: http://orca.cf.ac.uk/id/eprint/92124

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