Rapidly decaying solutions of the nonlinear Schrödinger equation

Abstract

We consider global solutions of the nonlinear Schrödinger equation

$$iu_t + \Delta u = \lambda |u|^\alpha u, in R^N ,$$

((NLS))

where λ∈R and$0< \alpha< \frac{4}{{N - 2}}$. In particular, for$\alpha _0 = \frac{{2 - N + \sqrt {N^2 + 12N + 4} }}{{2N}}$, we show that for every φεH ¹ (R ^N) such thatxφ(x)∈L ²(R ^N), the solution of (NLS) with initial value φ(x)e ^i(b|x|2/4) is global and rapidly decaying ast→∞ ifb is large enough. Furthermore, by applying the pseudo-conformal transformation and studying the resulting nonautonomous nonlinear Schrödinger equation, we obtain both new results and simpler proofs of some known results concerning the scattering theory. In particular, we construct the wave operators for$\frac{4}{{N + 2}}< \alpha< \frac{4}{{N - 2}}$. Also, we establish a low energy scattering theory for the same range of α and show that, at least for λ<0, the lower bound on α is optimal. Finally, if λ>0, we prove asymptotic completeness for$\alpha _0 \leqq \alpha< \frac{4}{{N - 2}}$.