Global well-posedness in Sobolev space implies global existence for weighted L² initial data for L² -critical NLS

الأصل · EN

The L² -critical defocusing nonlinear Schrodinger initial value problem on Rᵈ is known to be locally well-posed for initial data in L². Hamiltonian conservation and the pseudoconformal transformation show that global well-posedness holds for initial data u₀ in Sobolev H¹ and for data in the weighted space (1+|x|) u₀ in L². For the d=2 problem, it is known that global existence holds for data in Hˢ and also for data in the weighted space (1+|x|)σ u₀ in L² for certain s, σ< 1. We prove: If global well-posedness holds in Hˢ then global existence and scattering holds for initial data in the weighted space with σ= s.