By A. Jaffe (Chief Editor)
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Additional info for Communications in Mathematical Physics - Volume 184
A linear map l: E → F between locally convex spaces (lcvs) is smooth iff l is bounded. (ii) A bilinear map b: E × F → G of lcvs is smooth iff b is bounded. (i) One of the main questions at the starting point of the calculus is to provide convenient methods for testing smoothness of curves in order to check smoothness of more general maps. In Rn one can always look at the components of a “time-dependent” vector c(t) and differentiability is equivalent to differentiability of each component function.
N,N (n))T , ΨN (n) = det[1N + CN (n)] det[1N + CN (n − 1)] 1 2 0 T ΨN (n) = (c1 κn1 , . . , cN κN 1 ) , 1N + CN (n) −1 0 ΨN (n), n ∈ Z. 7), bN also admits the trace formula representation , bN (n) = − N 1 2 2 (κj − κ−1 j )ψN,j (n) , n ∈ Z. 20) satisfying HN fN,+ (k) = zfN,+ (k), in the weak sense. 21) 2 32 F. Gesztesy, W. 24) (note that TN (k)−1 = TN (k −1 )). 25) RN (with Rr (·), Rl (·) denoting the reflection coefficient from right and left incidence, respectively) and hence yield the following unitary scattering matrix SN (λ) in C2 associated with the pair (HN , H0 ), H0 = 21 (S + + S − ), SN (λ) = TN (k) 0 We also note that 0 , TN (k) λ= 1 (k + k −1 ) ∈ [−1, 1].
28) Finally, we briefly consider N –soliton solutions of the Toda lattice (in Flaschka’s variables) given by d a(t, n) = a(t, n)[b(t, n) − b(t, n + 1)], dt d b(t, n) = 2[a(t, n − 1)2 − a(t, n)2 )], dt (t, n) ∈ R × Z. 29). 29) then proves that HN (t) is unitarily equivalent to HN (0) for all t ∈ R. New Classes of Toda Soliton Solutions 33 3. Convergence Results as N → ∞ This is the main technical section in which we investigate various limits of aN (n), bN (n), HN , CN (n), fN,+ (k, n), ψN,− (k, n), and TN (k) as N → ∞.
Communications in Mathematical Physics - Volume 184 by A. Jaffe (Chief Editor)