  Let $T$ be the generator of a $C_0$-semigroup $e^{-Tt}$ which is of finite
trace for all $t>0$ (a Gibbs semigroup). Let $A$ be another closed operator,
$T$-bounded with $T$-bound equal to zero. In general $T+A$ might not be the
generator of a Gibbs semigroup. In the first half of this paper we give
sufficient conditions on $A$ so that $T+A$ is the generator of a Gibbs
semigroup. We determine these conditions in terms of the convergence of the
Dyson-Phillips expansion corresponding to the perturbed semigroup in suitable
Schatten-von Neumann norms.
  In the second half of the paper we consider
$T=H_\vartheta=-e^{-i\vartheta}\partial_x^2+e^{i\vartheta}x^2$, the
non-selfadjoint harmonic oscillator, on $L^2(\mathbb{R})$ and $A=V$, a locally
integrable potential growing like $|x|^{\alpha}$ for $0\leq \alpha<2$ at
infinity. We establish that the Dyson-Phillips expansion converges in this case
in an $r$ Schatten-von Neumann norm for $r>\frac{4}{2-\alpha}$ and show that
$H_\vartheta+V$ is the generator of a Gibbs semigroup
$\mathrm{e}^{-(H_\vartheta+V)\tau}$ for $|\arg{\tau}|\leq
\frac{\pi}{2}-|\vartheta|$. From this we determine asymptotics for the
eigenvalues and for the resolvent norm of $H_\vartheta+V$.
