Communications in Mathematical Physics - Volume 247 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Extra resources for Communications in Mathematical Physics - Volume 247

Sample text

As p and q run over s and sn+ , respectively, for 1 ≤ ≤ n−1, the sum p1 +· · ·+pn−1 −q1 −· · ·−qn−1 runs over a small set centered on some point k. In order for (s1 , · · · , s2n ) to be in Mom1 (s ), there must exist p ∈ sn ∩F and q ∈ s2n ∩F with q −p very close to k. But q −p is a secant joining two points of the Fermi curve F . We have assumed that F is strictly convex. 2n−3 q q k p p Consequently, for any given k = 0 in R2 there exist at most two pairs (p , q ) ∈ F 2 with q − p = k. So, if k is not near the origin, sn and s2n are almost uniquely determined.

F orm Proof. Let K ∈ Kj +1 and set K = renj,j +1 (K , W, u). dµC [j +1,j ) (ψ) u (K )¯ dµC [j +1,j ) (ψ) u (K )¯ 40 J. Feldman, H. Kn¨orrer, E. Trubowitz a) We first verify that (W , G , u ) is a formal interaction triple. 4 that is not trivially satisfied is uˇ (0, k); K = uˇ (0, k); K K ; q0 (K ) + qˇ0 ((0, k); K )ν (≥j +1) (0, k) ˇ ˇ = −K(k; K ; q0 (K )) + δ K(k; K ; q0 (K )) = −Kˇ (k). 8. 11. 8 is trivially fulfilled. 13) . Set U(K ) = 21 dξ1 dξ2 q0 (ξ1 , ξ2 ; K ):ψ(ξ1 )ψ(ξ2 ):Cj +1(u ;K ) .

3. For a function f on B m ×B n we define the (scalar valued) L1 –L∞ –norm as   max sup dξ |f (ξ1 , · · · , ξn )| ifm = 0  1≤j0 ≤n ξj ∈B j =1,··· ,n j 0 j =j0 |||f |||1,∞ =   sup dξj |f (η1 , · · · , ηm ; ξ1 , · · · , ξn )| if m = 0  η1 ,··· ,ηm ∈B j =1,··· ,n and the (d + 1)–dimensional L1 –L∞ seminorm f 1,∞ =     δ∈N0 ×Nd0    1 δ! max D decay operator with δ(D)=δ |||D f |||1,∞ t δ ifm = 0 . |||f |||1,∞ if m = 0 Here |||f |||1,∞ stands for the formal power series with constant coefficient |||f |||1,∞ and all other coefficients zero and dξ g(ξ ) = a∈{0,1} σ ∈{↑,↓} dx0 dx g (x0 , x, σ, a) .