Download Communications in Mathematical Physics - Volume 249 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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This class of Hamiltonians has been introduced and studied in various degrees of generality in [DG1, DJ, G1]. Pauli-Fierz Hamiltonians are defined in Subsect. 1 in an abstract framework. We establish there essentially an optimal condition under which the form φ(v) is small with respect to H0 , in form or operator sense, improving those which have been isolated in [G1]. Subsect. 2 is devoted to the study of the essential spectrum of these operators when φ(v) is a form perturbation of the free Hamiltonian.

20) and 1 ω ≤ Cb, |(u2 , [ω, wt ]u1 )| ≤ Ct b 2 u1 1 1 b 2 u2 , ui ∈ D(b 2 ), 0 < t < 1. 21) Then H0 ∈ C 1 (A; G, G ∗ ) and [H0 , iA]0 = IK ⊗ d ([ω, ia]0 ). 20) and 1 1 1 v ∈ B(D(K 2 ), K ⊗ D(a)), av ∈ B(D(K 2 ), K ⊗ D(b− 2 )). 22) Then φ(v) ∈ C 1 (A; G, G ∗ ) and [φ(v), iA]0 = −φ(iav). 22). Then H ∈ C 1 (A; G, G ∗ ) and [H, iA]0 = IK ⊗ d ([ω, ia]0 ) − φ(iav). 12. 21) implies that ω ∈ C 1 (a; D(b 2 ), D(b 2 )∗ ) and is equiva1 1 lent to it if b ≥ c > 0. Therefore [ω, ia]0 ∈ B(D(b 2 ), D(b 2 )∗ ) is well defined.

4 we have H = H0 + φ(v) as an operator sum in B(G, G ∗ ). We shall first prove i) assuming only that {wt } is a C0 -semigroup of contractions (so in our case the argument works both for {wt } and {wt∗ } ). Since Wt does not act on K we can without loss of generality assume that K = C and K = 0. We observe that fin (D(b)) is a form core for d (b). Then for u ∈ fin (D(b)): (Wt u, d (b)Wt u) = (u, Wt∗ d (b)Wt u) = (u, d (wt∗ wt , wt∗ bwt )u), which is less than Ct (u, d (b)u) because wt∗ wt ≤ I, wt∗ bwt ≤ Ct b.

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