Inverse problems and sharp eigenvalue asymptotics for Euler-Bernoulli operators

摘要

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.