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 $T=H_{\vartheta }=-e^{-i\vartheta }{\mathrm{\partial }}_x^2+e^{i\vartheta }{x}^2$, the non-selfadjoint harmonic oscillator, on $L^2(\mathbb{R})$ and $A=V$, a locally integrable potential growing like $\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }$ for $0\le \alpha <2$ 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 $H_{\vartheta }+V$.