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.

A regularized trace formula and oscillation of eigenfunctions of a Sturm‐Liouville operator with retarded argument at 2 points of discontinuity

“…Thus, when the number of points of discontinuity is more than 1, we see how the behavior of trace and oscillations of eigenfunctions of a boundary value problem with retarded argument which contains a spectral parameter in the boundary conditions change. We point out that our results are extension and/or generalization to those in Gelfand and Levitan, Pikula, Yang, Hira, Yurko, Dikii, Shieh and Yurko, and Cheng and Law . For example, if the retardation function Δ≡0 in , , δ =1, and γ =1, we have the formula of the first regularized trace for the classical Sturm‐Liouville operator which is called Gelfand‐Levitan formula (see Gelfand and Levitan).…”

Solutions to open problems of Yang concerning inverse nodal problems

“…In particular, Lemma 2.1 (II) corrects the corresponding statements of Lemma 3.1 in [19], where λ n is expressed in terms of n, but not n − 1. The confusion stems from a basic counting lemma which should state | √ λ n − (n − 1)π| < π/2 for large n (see [5,16]). The detailed discussion depends on the boundary conditions.…”

“…Inverse nodal problems for classical Sturm-Liouville operators have been studied fairly completely. The main results in this area are presented in the monographs [1,2,3,4,5,6,7,9,10,12,13,14,15,17,19,20] and other works.…”

The inverse nodal problem for Hill's equation

“…Note that the eigenvalue λ n can be replaced by its asymptotic expansion involving 1 0 q only so that the right-hand side above is a sequence of step functions depending only on the nodal data plus the constant 1 0 q. Recently we, extending the result in [13], showed that ( [4]) the space of Sturm-Liouville operators characterized by H = (q, α, β) ∈ L 1 (0, 1) × [0, π) 2 such that 1 0 q = 0 is homeomorphic to the partition set of the space of quasinodal sequences, i.e. the space of all admissible sequences X = X (n) k which form sequences that converge to q, α and β individually.…”