Uniqueness of solutions to the 3D quintic Gross–Pitaevskii hierarchy

By Younghun Hong, Kenneth Taliaferro, Zhihui Xie

http://www.sciencedirect.com/science/article/pii/S0022123615004115

In this paper, we research suggestions to the 3-dimensional quintic Gross–Pitaevskii hierarchy. We end up unconditional forte between all small strategies within the severe area H1 (which corresponds to H1 at the NLS level). With moderate ameliorations to the evidence, we additionally end up unconditional forte of recommendations to the Hartree hierarchy and not using a smallness . Our evidence makes use of the quantum de Finetti theorem, and is an extension of the paintings by way of Chen, Hainzl, Pavlović, and Seiringer, and our earlier paintings.

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16)) takes values σ1 (2) = 1, σ1 (4) = three, and σ1 (6) = three. For j = 2, we practice the relabeling within the similar spirit find that J21 = U0,2 B1;2,3 U2,4 (|φ φ|)⊗3 , (1) (3) (3. nine) the place σ2 (2) = 1. We word that for any < , the interplay operators Bσ( ); , +1 and Bσ( ); , +1 in J 2 (which are highlighted in (3. 1)) belong to an identical issue Jj1 if both σ( ) = σ( ) or σ( ) = . In such situations, we think about them as being hooked up. This connectivity constitution is strictly the main element of the Duhamel phrases that we wish to illustrate utilizing tree graphs.

2. 1. Setup (k) (k) ˙1 enable {γ1 (t)}k∈N and {γ2 (t)}k∈N be light recommendations in L∞ t∈[0,T ) H that resolve (1. 7) with an analogous preliminary information, and are both admissible or restricting hierarchies. To end up distinctiveness, we'll convey that their difference {γ (k) (t)}k∈N , given via (k) (k) γ (k) (t) := γ1 (t) − γ2 (t), okay ∈ N, (2. 1) is 0. certainly, it suffices to end up the next. Proposition 2. 1. For A > 1 > a > zero, we define the medium frequency cut-offs through Pa,A,j = 1{a≤−Δxj ≤A} and Pa,A,j = 1{a≤−Δx ≤A} , the place 1a≤|x|≤A = 1 if a ≤ |x| ≤ A and j 1a≤|ξ|≤A = zero another way.

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For that reason, J 2 will be factorized as (1) (1) J 2 = (U0,1 Θ1 ) ⊗ (U0,2 Θ2 ) := J11 ⊗ J21 . (3. 7) Now J 2 in (3. 7) has elements Jj1 (note j ≤ okay = 2), that are 1-particle matrices. the explanation we've got this sort of decomposition is that Bσ1 (r);r,r+1 simply affects 3 ui whenever, and because the contraction strategies, the entire ui could be divided into different teams. For j = 1, after exchanging again ui = |φ φ|, i ≤ okay + 2n = 10, we've J11 = U0,1 B1;2,3 U1,3 B3;4,5 U3,4 B3;6,7 (|φ φ|)⊗7 (1) (2) (3) (3. eight) the place we relabel the index in operators Bσ1 (r);r,r+1 such that the interplay operators in (3.

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