  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.
