# Resonances for Euler-Bernoulli operator on the half-line

## TL;DR

This paper studies the resonances of the Euler-Bernoulli operator on the half-line, providing asymptotic counts, forbidden regions, trace formulas, and conditions for the absence of resonances and eigenvalues.

## Contribution

It offers new asymptotic and trace formula results for fourth order operators and characterizes when the Euler-Bernoulli operator has no resonances or eigenvalues.

## Key findings

- Resonance counting asymptotics at large radius
- Forbidden domain for resonances identified
- Resonances absent iff coefficients are constant

## Abstract

We consider resonances for fourth order differential operators on the half-line with compactly supported coefficients. We determine asymptotics of a counting function of resonances in complex discs at large radius, describe the forbidden domain for resonances and obtain trace formulas in terms of resonances. We apply these results to the Euler-Bernoulli operator on the half-line. The coefficients of this operator are positive and constants outside a finite interval. We show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half-line.