# Download Communications in Mathematical Physics - Volume 260 by M. Aizenman (Chief Editor) PDF

By M. Aizenman (Chief Editor)

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Extra info for Communications in Mathematical Physics - Volume 260

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Ar ; q, p)∞ = (a1 ; q, p)∞ · · · (ar ; q, p)∞ . For a function f = f (x), we let the symbols f and f stand for its qshifts: f = f (qx), f = f (q −1 x). 1. 1. Definition. We first recall the definition of the universal character. For a pair of sequences of integers λ = (λ1 , λ2 , . . , λl ) and µ = (µ1 , µ2 , . . , µl ), the universal character S[λ,µ] (x, y) is a polynomial in (x, y) = (x1 , x2 , . . , y1 , y2 , . . 1) 1≤i,j ≤l+l where pn are defined by the generating function: ∞ pk (x)zk = exp k∈Z xn zn .

Phys. B: At. Mol. Opt. Phys. 34, L113–L119 (2001) 2. : An instability property of the nonlinear Schrodinger equation on Sd. Math. Res. Let. 9, 323–335 (2002) 3. : Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations. Amer. J. Math. 125, 1235–1293 (2003) 4. : Spectral Asymptotics in the semi-classical limit. Cambridge: Cambridge University Press, 1999 5. : Feshbach resonances and collapsing Bose-Einstein condensates. New J. Phys. 11 (2003) 6. : Bose-Einstein Condensation.

Throughout this paper, we shall use the following notation of q-shifted factorials: ∞ (a; q)∞ = ∞ (1 − q i a), (a; q, p)∞ = (1 − q i p j a); i,j =0 i=0 also (a1 , . . , ar ; q)∞ = (a1 ; q)∞ · · · (ar ; q)∞ and (a1 , . . , ar ; q, p)∞ = (a1 ; q, p)∞ · · · (ar ; q, p)∞ . For a function f = f (x), we let the symbols f and f stand for its qshifts: f = f (qx), f = f (q −1 x). 1. 1. Definition. We first recall the definition of the universal character. For a pair of sequences of integers λ = (λ1 , λ2 , .