## Abstract

In this paper we consider analyticity in time and smoothing effect of solutions to nonlinear Schrödinger equations {i∂_{t}u + 1/2△u = λ|u|^{2p}u, (t, x) ∈ ℝ × ℝ^{n}, u(0, x) = φ, x ∈ ℝ^{n}, (1) where λ ∈ ℂ, p ∈ ℕ. We prove that if φ satisfies ∥e^{|x|2}φ∥ _{Η[n/2]+1} < ∞, (2) then there exists a unique solution u(t, x) of (1) and positive constants Τ, C_{0}, C_{1} such that u(t, x) is analytic in time and space variables for t ∈ [-Τ,Τ] \ {0} and x ∈ Ω = {x; |x| < R} and has an analytic continuation U(z_{0}, z) on {z_{0} = t + iτ; -C_{0}t^{2} < τ < C_{0}t^{2},t ∈ [-Τ,Τ] \ {0}} and {z = x + iy; -C_{1}|t| < y < C_{1}(t),(t, x) ∈ [-Τ,Τ] \ {0} × Ω} . In the case n = 1,2,3 the condition (2) can be relaxed as follows: ∥e^{|x|2}φ∥ _{Ηm} < ∞, where m = 0 if n = 1, p = 1, m = 1 if n = 2. p ∈ ℕ and m = 1 if n = 3, p = 1.

Original language | English |
---|---|

Pages (from-to) | 273-300 |

Number of pages | 28 |

Journal | Communications in Mathematical Physics |

Volume | 184 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1997 |