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

Malcolm Brown,Marco Marletta,Freddy Symons

doi:10.1112/jlms/jdw059

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.

Highlights

We prove a pair of uniqueness theorems for an inverse problem for an ordinary differential operator pencil of second order
We prove a limit-circle analogue to the limit-point m-function interpolation result of Rybkin and Tuan
In the space H we examine the following operator pencil: Lu(r; λ) = λP u(r; λ) (r ∈ (0, 1))

Summary

Introduction: new definitions and problem statements

In the PLP case ν 2 (note ν = 2 turns out to be treatable by a PLP technique), we will transform (1.3) to Liouville normal form on the half-line [0, ∞), in PLP at ∞, regular at 0, before utilizing the Rybkin–Tuan interpolation formula [24] for the classical limit-point m-function associated with such an equation. This is valid because the PLP and classical limitpoint m-functions are formally the same where their domains overlap, that is, all of C when ν > 2 and Ωp when ν = 2; the Rybkin–Tuan interpolation holds in this region. We will conclude the paper with an illustration of the relevance of this result in Section 4, where we explain how it proves a uniqueness theorem for the physically motivated Berry– Dennis PDE inverse problem, which involves boundary singularities and partial Cauchy data at the boundary

Interpolation of a classical limit-circle m-function

Uniqueness theorems for the inverse problem

Findings

The Berry–Dennis problem