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 trace class 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 in suitable Schatten-von Neumann norms. In the second half of the paper we consider T=H _ϑ=−e ^−iϑ∂ _x ²+e ^iϑx ², the non-selfadjoint harmonic oscillator, on L ²(R) and A=V, a locally integrable potential growing like |x| ^α at infinity for 0≤α<2. We establish that the Dyson-Phillips expansion converges in r Schatten-von Neumann norm in this case for r large enough and show that H _ϑ+V is the generator of a Gibbs semigroup e ^{−(H _ϑ+V)τ} for |arg⁡τ|≤ [Formula presented] −|ϑ|≠ [Formula presented]. From this we determine high energy asymptotics for the eigenvalues and the resolvent norm of H _ϑ+V.