2. Solving the linear Schrödinger equation🔗

We start by considering the initial value problem for the linear Schrödinger equation on \mathbb{R}^n,

\left\{\begin{aligned}i∂_t u(t, x) &= Δ u \\ u(0, x) &= u₀.\end{aligned}\right.

We want to prove that for sufficiently regular u₀ there exists a unique solution u solving the initial value problem and u has some regularity.

1. 2.1. Well-posedness in Schwartz functions
2. 2.2. Well-posedness in tempered distributions
3. 2.3. Well-posedness on Sobolev space
4. 2.4. Explicit representation of the solution operator