# Communications In Mathematical Physics - Volume 277 by M. Aizenman (Chief Editor) PDF By M. Aizenman (Chief Editor)

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Fourth step: concluding the proof of Proposition 2. 19), with ⎛ ⎞ q m φ ⎝ε φ(ki ) F(k1 , . . 30) : j ∈E p we readily see that the proposition is proved once N (F) is conveniently estimated. We first observe, setting G A (K ) = G(K )ei A·K , where A ∈ Rdq and K = (k1 , . . , kq ), that N (G A ) = N (G). 31) This follows by an elementary computation. 32) i=1 m ’s. 31), we thus obtain q m N (F) ≤ dY |φ(yi )| N i=2 Rd(m−1) m−1 q ≤ φ L 1 c N (φ)q e−ibK ·Y φ(ki ) i=1 ≤ c N (φ) . n n In conclusion, we have established n T (t, ξ1 , η1 ) ≤ ε 2 −m+1 t m−1 cn (m − 1)!

Proof of Lemma 5. It is enough to consider the case when i |ai | > 0. We first estimate dz dz p ≤ ∗ k k |ai | + ∗ |ai z + xi | |ai | + |ai z + xi | R i=1 where the symbol ∗ designates . Next, by convexity, we may write i:ai =0 ∗ ∗ |ai z + xi | = p, R i=1 |ai | z + ⎞ ⎛ ⎜ ⎜ ⎜z+ ⎝ ∗ xi ≥ |ai | |ai | ∗ xi |ai | ∗ ⎟ ⎟ ⎟. ⎠ =:x¯ The change of variables z + x¯ → z in the above integral finishes the proof. By the Parseval formula and the explicit form of the Fourier transform of a complex Gaussian, we have q dk j F(k1 , .

The linear boltzmann equation as the low density limit of a random schrödinger equation. Rev. Math. Phys. : Linear Boltzmann Equation as Scaling Limit of Quantum Lorentz Gas. In: Advances in differential equations and mathematical physics (Atlanta, GA, 1997), Contemp. Math. 217, Providence, RI: Amer. Math. , 1998, pp. : Linear boltzmann equation as the weak coupling limit of a random schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000) 44 [ESY] [H] [HL] [IP] [L] [KP] [LP] [MRS] [RS] [RV] [Sp] [U] [UU] [W] D.