2.1. Well-posedness in Schwartz functions🔗

The best possible class of initial data is if u₀ ∈ 𝓢(ℝ^n). Then we can use the Fourier transform to obtain a candidate for the solution as

u(t, x) = 𝓕⁻ \left( e ^ {-i t ‖ξ‖ ^ 2 / (4 π ^ 2)} (𝓕 u₀)(ξ) \right)

We claim that u ∈ C^∞(ℝ, 𝓢(ℝ^n)) and u solves the Schrödinger equation with initial data u₀.