  The notion of equivalence classes of generators of one-parameter semigroups
based on the convergence of the Dyson expansion can be traced back to the
seminal work of Hille and Phillips, who in Chapter XIII of the 1957 edition of
their Functional Analysis monograph, developed the theory in minute detail.
Following their approach of regarding the Dyson expansion as a central object,
in the first part of this paper we examine a general framework for perturbation
of generators relative to the Schatten-von Neumann ideals on Hilbert spaces.
This allows us to develop a graded family of equivalence relations on
generators, which refine the classical notion and provide
stronger-than-expected properties of convergence for the tail of the
perturbation series. We then show how this framework realises in the context of
non-self-adjoint Schrodinger operators.
