Inverse problems and sharp eigenvalue asymptotics for Euler-Bernoulli operators

TL;DR

This paper investigates inverse problems and eigenvalue asymptotics for Euler-Bernoulli operators, providing new sharp asymptotic formulas and an Ambarzumyan-type theorem for operators with real coefficients on the unit interval.

Contribution

It introduces a novel Ambarzumyan-type theorem and derives sharp eigenvalue asymptotics for Euler-Bernoulli operators with converging coefficients and at high energy.

Findings

Established an Ambarzumyan-type theorem for inverse problems.

Derived sharp eigenvalue asymptotics for operators with converging coefficients.

Extended eigenvalue asymptotics to complex coefficient operators at high energy.

Abstract

We consider Euler-Bernoulli operators with real coefficients on the unit interval. We prove the following results: i) 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 both for the Euler-Bernoulli operator and fourth order operators (with complex coefficients) on the unit interval at high energy.