An inverse spectral problem for the Euler - Bernoulli equation for the vibrating beam

Abstract

The equation of the vibrating beam, together with some appropriate boundary conditions, can be viewed as an eigenvalue problem for the operator L given by Lu = rho(-1)(au '')'' where the functions a and rho are strictly positive, and correspond to physical characteristics of the beam. In this work we examine how a given spectral datum determines whether a(x)rho(x) = 1. If this equation holds, L becomes a 'perfect square'.