Spectral theory of time-periodic many-body systems

Commutators, C0-semigroups and resolvent estimates

“…Since the operator A will be assumed maximal symmetric, it will be clear that the abstract theory developed in [22] is also a particular case of ours. Later on in this introduction we shall explain why the results of [32] are consequences of ours and we shall discuss the relation with [25].…”

“…We just mention here that they are an ingredient in the derivation of the Fermi Golden Rule in second-order perturbation theory of embedded eigenvalues, as presented in [1]. See also [12,25]. Higher orders of regularity have been used recently by Agmon and Herbst [2] to make a precise study of perturbations under which an embedded eigenvalue persists.…”

“…We shall make now some comments concerning the relation between our paper and [25,32]. The assumptions of [32] involve, besides the self-adjoint operator H and the maximal symmetric operator A; an auxiliary self-adjoint operator MX1 (in applications this is the particle number operator plus the projection onto the vacuum state) and a sequence of self-adjoint operators A n which converge to A: Most of Skibsted's hypotheses are formulated in terms of the commutators between H and A n and are similar to those of Mourre [26] with one notable exception: ½H; iA n does not satisfy the Mourre positivity condition.…”

“…In papers [25,32] the family H e is taken to be H MS e ¼ H À ieðM þ f ðHÞGf ðHÞÞ: This choice differs from ours (namely H e ¼ H À ieH 0 ) due to the appearance of the energy cutoffs f ðHÞ: In [25] it is observed that H and M appear symmetrically in H MS e ; up to the need for uniformity of estimates in e: This observation makes it possible to ease the assumption on ½H; iM mentioned above, such as to cover the application considered there. An assumption of the following form is introduced…”

“…Moreover, the domination conditions were weakened by various authors, which increased the power and elegance of the theory (see [4] and references therein). Among the various extensions of the Mourre theorem which exist in the literature, two are especially interesting for us: the first is due to Hu¨bner and Spohn [22] and the second to Skibsted [32] (this is further developed in [25]). In [22] it is shown that Mourre's results remain true if A is only maximal symmetric, the main technical result being the extension of the virial theorem to this context (the resolvent estimates extend easily because domination conditions similar to those of Mourre are imposed).…”