Cardiff University | Prifysgol Caerdydd ORCA
Online Research @ Cardiff 
WelshClear Cookie - decide language by browser settings

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

Brown, Malcolm, Marletta, Marco and Symons, Freddy 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

[thumbnail of 151027_articleclass_Uniqueness_Brown_Marletta_Symons.pdf.pdf]
PDF - Published Version
Available under License Creative Commons Attribution.

Download (326kB) | Preview


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: Mathematics
Computer Science & Informatics
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Additional Information: This is an Open Access article distributed under the terms of the Creative Commons Attribution License (, which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
Publisher: Wiley
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: 26 Feb 2020 16:30

Actions (repository staff only)

Edit Item Edit Item


Downloads per month over past year

View more statistics