4

For the sake of concreteness, we consider the linear Schrodinger equation $$ \partial_t u = i\Delta u, \ \ \ \ u(0, x) = u_0(x). $$ The solution is typically (at least, how I've seen it) 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), dominated convergence 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 Schrodinger 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 $\partial_t \hat{u} \neq \widehat{\partial_t u}$?

Any help is much appreciated!

Chris
  • 5,231
  • Hmm, I can easily find examples of problematic functions when swapping two derivatives, but what about swapping a derivative and an integral (which FT is AFAIK). I could not find any info on this phenomenon by googling? Do you know of a single example of such a function? – Aleksejs Fomins Nov 13 '19 at 10:10
  • In general as far as I know, we need to justify swapping derivatives and integrals via the Leibniz rule: https://en.wikipedia.org/wiki/Leibniz_integral_rule. At the end the most general statement of it involves the dominated convergence theorem. – Chris Nov 14 '19 at 00:23

1 Answers1

1

Resolved on mathoverflow: uniqueness can in fact fail if sufficient decay/smoothness is not enforced on the solution (see Tao, Exercise 2.24). In particular, we get uniqueness of solutions in $C_t^1\mathcal{S}_x$.

Chris
  • 5,231
  • 2
    That was a good question. The conditional uniqueness is a standard feature of PDEs, the standard example being Tychonoff's one of a heat solution $u$ that is everywhere positive but $u|_{t=0}=0$. – Giuseppe Negro Nov 14 '19 at 20:28
  • 2
    Continuing my previous comment; see Evans, "Partial differential equations", 2nd edition, pag. 59: "Nonphysical solutions" (remark after Theorem 7). – Giuseppe Negro Nov 16 '19 at 15:45