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

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–Marčenko argument.