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.

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