By Pauling L.

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Here we recall only a few basic results concerning thermodynamics in the weak coupling limit (for additional Topics in Non-Equilibrium Quantum Statistical Mechanics 23 information see [LeSp]). We will assume that the general conditions described in the lecture notes [D1] are satisfied. The operator KH generates a positivity preserving contraction semigroup on OS . Obviously, KH (I) = 0. We will assume that zero is the only purely imaginary eigenvalue of KH and that Ker KH = CI. This non-degeneracy condition can be naturally characterized in algebraic terms, see [D1,Sp].

Then for A ∈ O0 and B ∈ O0+ (34) holds. If h0 is dense in h, then O0 is dense in CAR(h) and O0+ is dense in CAR+ (h). Let h1 and h2 be two Hilbert spaces, and let Ωh1 , Ωh2 be the vaccua in Γ− (h1 ) and Γ− (h2 ). 20) states that there exists a unique unitary map U : Γ− (h1 ⊕ h2 ) → Γ− (h1 ) ⊗ Γ− (h2 ) such that U Ωh1 ⊕h2 = Ωh1 ⊗ Ωh2 , U a(f ⊕ g)U −1 = a(f ) ⊗ I + (−I)N ⊗ a(g), U a∗ (f ⊕ g)U −1 = a∗ (f ) ⊗ I + (−I)N ⊗ a∗ (g), (35) U dΓ (h1 ⊕ h2 )U −1 = dΓ (h1 ) ⊗ I + I ⊗ dΓ (h2 ). The presence of the factors (−I)N in the above formulas complicates the description of a system containing several reservoirs.

2 2 λ→0 (60) (ii) For any t ≥ 0 and j = 1, · · · , M , 2 2 lim (pj eithλ /λ 1, TRj pj eithλ /λ 1) = λ→0 2 |fj (ε0 )|2 ρj (ε0 ) 1 − e−2πt|f (ε0 )| . 1 is not difficult—for Part (i) see [Da1, D1], and for Part (ii) [Da2]. These proofs use the regularity Assumption (SEBB5). 1, based on the explicit form of the wave operator W− , can be found in [JKP].