# Literature request: Schatten class difference of semigroups

Let $$\mathcal{H}$$ be a Hilbert space and $$A,B$$ two operators on it (not necessarily self-adjoint) such that $$A, A+B$$ are generators of strongly continuous one parameter semigroups $$e^{-tA},e^{-t(A+B)}$$, $$t>0$$. I would like to ask for some literature on the consequences of the condition $$$$e^{-t(A+B)}-e^{-tA} \in S_{p}(\mathcal{H})$$$$ where $$p\geq 1$$ and $$S_{p}(\mathcal{H})$$ is the p-Schatten class of operators on $$\mathcal{H}$$. In particular, I would like results crucially using that this difference is Hilbert-Schmidt, for example, and not just that it is compact as a result.

The application they give is $$e^{t(\Delta - V)} - e^{t\Delta}$$ on the whole space, where the semigroups are not even compact, but the difference belongs to the Schatten classes you wish for.