Perturbations of Gibbs semigroups and the non-selfadjoint harmonic oscillator

Abstract

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 \(H_\vartheta+V\).