2023/05/21 by Jesse Gell‐Redman, Gell-Redman, Jesse, Sean Gomes +3 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2305.12429
Using the Fredholm theory of the linear time-dependent Schrödinger equation set up in our previous article arXiv:2201.03140, we solve the final-state problem for the nonlinear Schrödinger problem (Dt + Δ+ V) u = N[u], u(z,t) ∼ (4πit)-n/2 ei|z|2/4t f( (z)/(2t) ), t → -∞, where u : ℝn+1 → ℂ is the unknown and f : ℝn → ℂ is the asymptotic data. Here Dt = -i (∂)/(∂ t) and Δ= ∑j=1n Dzj Dzj is the positive Laplacian, or more generally a compactly supported, nontrapping perturbation of this, V is a smooth compactly supported potential function, and the nonlinear term N is a (suitable) polynomial in u, ∂zju and their complex conjugates satisfying phase invariance. Our assumption on the asymptotic data f is that it is small in a certain function space Wk constructed in arXiv:2201.03140, for sufficiently large k ∈ ℕ, where the index k measures both regularity and decay at infinity (it is similar to, but not quite a standard weighted Sobolev space Hk, k(ℝn)). We find that for N[u] = ± |u|p-1 u, p odd, and (n,p) ≠ (1, 3) then if the asymptotic data as t → -∞ is small in Wk, then the asymptotic data as t → +∞ is also in Wk; that is, the nonlinear scattering map preserves these spaces of asymptotic data. For a more general nonlinearity involving derivatives of u, we show that if the asymptotic data as t → -∞ is small in ⟨ ζ⟩-1 Wkζ, then the asymptotic data as t → +∞ is also in this space (where ζ is the argument of f).