Inverse problems and sharp eigenvalue asymptotics for Euler–Bernoulli operators

Abstract

We consider Euler–Bernoulli operators with real coefficients on the unit interval. We prove the following results: (i) The Ambarzumyan-type theorem about the inverse problems for the Euler–Bernoulli operator. (ii) The sharp asymptotics of eigenvalues for the Euler–Bernoulli operator when its coefficients converge to the constant function. (iii) The sharp eigenvalue asymptotics for both the Euler–Bernoulli operator and fourth-order operators (with complex coefficients) on the unit interval at high energy.