# Resonances of 4-th Order Differential Operators

## TL;DR

This paper analyzes the resonances of fourth order differential operators, deriving asymptotics for their count at large radii and characterizing conditions for the absence of eigenvalues and resonances.

## Contribution

It provides new asymptotic formulas for the number of resonances and characterizes when Euler-Bernoulli operators have no eigenvalues or resonances.

## Key findings

- Asymptotics of resonance count at large radius are established.
- Euler-Bernoulli operators have no eigenvalues or resonances iff coefficients are globally constant.
- Resonance distribution is explicitly described for certain fourth order operators.

## Abstract

We consider fourth order ordinary differential operator with compactly supported coefficients on the line. We determine asymptotics of the number of resonances in complex discs at large radius. We consider resonances of an Euler-Bernoulli operator on the real line with the positive coefficients which are constants outside some finite interval. We show that the Euler-Bernoulli operator has no eigenvalues and resonances iff the positive coefficients are constants on the whole axis.