On the quasi-nodal map for the Sturm–Liouville problem

Abstract

We show that the space of Sturm–Liouville operators characterized by H = (q, α, β) ∈ L¹ (0, 1) × [0, π)² such that is homeomorphic to the partition set of the space of all admissible sequences which form sequences that converge to q, α, and β individually. This space, Γ, of quasi-nodal sequences is a superset of, and is more natural than, the space of asymptotically nodal sequences defined in Law and Tsay (On the well-posedness of the inverse nodal problem. Inv. Probl.17 (2001), 1493–1512). The definition of Γ relies on the L¹ convergence of the reconstruction formula for q by the exactly nodal sequence.