In a recent paper, X. F. Yang proved a uniqueness theorem on inverse nodal problems that links to inverse spectral theory, on one hand, and reduces the redundancy of the classical inverse nodal problems, on the other hand. In this note we improve Yang's theorem by weakening its conditions and simplifying its proof. We also discuss variants of Yang's theorem. ᮊ

We study the inverse nodal problem for Hill's equation. In particular, we solve the uniqueness, reconstruction and stability problems using the nodal set of periodic (or anti-periodic) eigenfunctions. Furthermore, we show that the space of periodic potential functions q normalized by ∫10q = 0 is homeomorphic to the partition set of the space of quasinodal sequences. Our method is to make a translation so that the periodic (or anti-periodic) problem is reduced to a Dirichlet problem.

Abstract. It is known that the potential function of the Sturm-Liouville problem can be reconstructed from the nodal data by a pointwise limit. We show that this convergence is in fact L 1 .

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a b s t r a c tIn this paper, we find the minimizer of the eigenvalue gap for the Schrödinger equation and vibrating string equation. In the first part, we show the first two Neumann eigenvalue gap of the Schrödinger equation with single-well potentials is not less than 1 and the equality holds if and only if the potential is constant. In the second part, since the first Neumann eigenvalue of the vibrating string equation is 0, we turn to show that the minimizing density function of the second Neumann eigenvalue is of the form hχ (a,π−a) +Hχ [0,π]\(a,π−a) for some a.