Ambarzumyan-type theorem for third order linear measure differential equations

Abstract

This paper deals with the Ambarzumyan-type theorem for a complex third order linear measure differential equation idy′•+2iqxy′dx+yidqx+dpx=λydx on [0, 1] with boundary conditions y1=0, y′1=y′0, and hy(0)+y′•0=0, where p∈M(I,R), q∈M0(I,R), and h=−h̄. More precisely, we prove that if the eigenvalues of this boundary value problem are (2nπ)3, n = 0, ±1, ±2, …, then h = 0 and the measure coefficients p(x) = p(0), q(x) = 0 for x ∈ [0, 1).