Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

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 $T=H_vartheta=-e^{-ivartheta}partial_x^2+e^{ivartheta}x^2$, the non-selfadjoint harmonic oscillator, on $L^2(mathbb{R})$ and $A=V$, a locally integrable potential growing like $|x|^{alpha}$ for $0leq alpha<2$ 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 $H_vartheta+V$.