# Third order operators with three-point conditions associated with Boussinesq's equation

## TL;DR

This paper studies a third order differential operator with specific boundary conditions linked to the Boussinesq equation, deriving eigenvalue asymptotics and trace formulas to aid inverse spectral analysis.

## Contribution

It introduces a detailed spectral analysis of a non-self-adjoint third order operator with three-point conditions related to the Boussinesq equation, including eigenvalue asymptotics and trace formulas.

## Key findings

- Eigenvalue asymptotics at high energy are established.
- Trace formula for the operator is derived.
- Eigenvalues form an auxiliary spectrum for inverse problems.

## Abstract

We consider a non-self-adjoint third order operator on the interval $[0,2]$ with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator.