Ambarzumyan theorem by zeros of eigenfunction

Abstract

In this short paper, we give the proof of the Ambarzumyan theorem by zeros of eigenfunctions (nodal points) different from eigenvalues for the one-dimensional p-Laplacian eigenvalue problem. We show that the potential function q(x) is zero if the spectrum is in the specific form. We consider this theorem for p-Laplacian equation with periodic and anti-periodic cases. If p = 0, results are coincided with the results given for Sturm-Liouvile problem.