Onthe well-posedness of the inverse nodal problem

Abstract

The inverse nodal problem on the Sturm-Liouville operator is theproblem of finding the potential function q and boundaryconditions α,β using the nodal sequence {xk(n)}.In this paper, we show that the space of all (q,α,β) suchthat ∫01q = 0, under a certain metric, is homeomorphic tothe partition set of all asymptotically equivalent nodalsequences induced by an equivalence relation. As a consequence,the inverse nodal problem, when defined on the partition set ofadmissible sequences induced by the same equivalence relation,is well posed. Let Φ be the homeomorphism, which we calla nodal map. We find that Φ is still a homeomorphismwhen the corresponding metrics are magnified by the derivativesof q, whenever q is CN. Our method depends heavily on theexplicit asymptotic expressions of the nodal points and nodallengths.