# Justification for uniqueness of solutions to dispersive PDE

For the sake of concreteness, we consider the linear Schrödinger equation $$\partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x).$$ The solution is typically obtained by taking the Fourier transform of both sides, giving $$\widehat{\partial_t u}(t, \xi) = -i|\xi|^2 \hat{u}(t, \xi)$$.

The next step is where I have questions. Assuming that everything is nice enough (for instance, in Tao's book, he assumes $$u_0$$ is Schwartz), a dominated convergence argument gives $$\widehat{\partial_t u}(t, \xi) = \partial_t \hat{u}(t, \xi)$$, and then we get an ODE that solves to $$\hat{u}(t, \xi) =e^{-i|\xi|^2}\hat{u}_0(\xi) \implies u(t, x) = e^{it\Delta}u_0(x).$$ This is then referred to as "the solution to the Schrödinger equation, with initial data $$u_0$$."

My question: How do we know that there are no other solutions, that may not satisfy the right decay/smoothness criteria to justify pulling the Fourier transform into the time derivative of $$u$$? I agree that there are no other solutions $$u$$ that are "nice enough" to justify this. But how do we rule out the existence of solutions $$u$$ such that $$\partial_t \hat{u} \neq \widehat{\partial_t u}$$? For instance, I don't understand how just assuming $$u_0$$ Schwartz is enough to guarantee this.

Any help is much appreciated.

• In my book I only establish uniqueness for solutions that are $C^1_t {\mathcal S}_x$ by this argument. Without some regularity in time and decay in space uniqueness can fail; see Exercise 2.24 of the book. Nov 13, 2019 at 5:54