Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?

Question

I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the function $f$ of real variables $t$ and $x$, it is asked whether any solution exists, defined for all $x$ and in a neighborhood of $t=0$, meeting the following conditions (restated following Terry Tao answer):

1. $f(x,t)$ is infinitely differentiable in $x$ and $t$, and
2. $f(x,t)$ is compactly supported in $x$ for every $t$,

i.e., do exist any compactly supported $C^\infty$ initial condition $f_0(x) \equiv f(x, 0)$ that evolves into a compactly supported solution in a neighborhood of $t=0$?

I expect an answer in the no, according to the following intuition:

• $f(x,t)$ can be obtained by convolution of $f_0(x)$, in the variable $x$, with the kernel (propagator) $K(x,t)=\left(2 \pi\, t\right)^{-1/2} \exp( i\, x^2/(2 t)- i \pi/4)$. Inspection of easily obtainable closed form solutions for compaclty supported and regular (albeit not $C^\infty$) $f_0$, as well as asymptotic analysis for $ t \to 0$ suggest that $f$ is bound to oscillate rapidly, whilst tailing off to the $x$ infinity, for any $t\ne 0$.
• In Fourier transform, the compact support of $f_0(x)$ should prevent the entire funcion $f_0(k)$ from falling at infinity rapidly enough to temper the exponential term in the transformed solution $f(k, t) = \exp(-i\,t\,k^2/2) f_0(k)$, thus contradicting for $t\ne 0$ the Schwartz condition $|f(k,t)| < A_n (1+|k|)^{-n} \exp(-B | \mbox{Im}( k)|)$ met by $f(k,t)$ in case $f(x,t)$ has compact support (in general, references on the precise characterization of the Fourier transform of the space of $C^\infty$ compactly supported functions, in terms of both upper and lower bounds on their asymptotic behavior in the complex plane, would be appreciated).

If so, and as a motivation for asking, any $C^\infty$ non-zero solution $f(x,t)$ that is compactly supported at $t=0$ has non-zero values that reach the $x$ infinity for any $t>0$ – a fairly curious and counterintuitive conclusion (to me) both mathematically, since a step by step integration in the variable $t$ should fail for any compactly supported inital condition $f_0$, and physically, since a free particle - as idealized by the Schrodinger equation - should be impossible to confine, in principle, for a finite time interval within a finite region of space (in practice of course, the far reaching tails of the wavefunction would quickly fade beyond detecdability).

Thanks in advance for your answers

Answer 1

I think the question you stated is not the one you intended, since the Schrodinger evolution is indeed a smooth flow in the Schwartz class, as can be seen by Fourier analysis, and so smooth compactly supported data $f_0$ always leads to a smooth global solution which is also Schwartz in space.

My guess is that you are asking instead to exclude the possibility of a nontrivial solution which is compactly supported at more than one time (not just at the initial time $t=0$). This was recently established (in quite great generality) in a series of papers by Escuriaza, Kenig, Ponce and Vega: see e.g. the survey http://arxiv.org/abs/1110.4873 . The methods are related to those used to establish unique continuation for elliptic operators, e.g. Carleman type estimates. There is also a close connection with the Hardy uncertainty principle, which ties in with your intuition that the relationship between the decay of a function and of its Fourier transform should be relevant.