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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

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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.

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  • $\begingroup$ @Dear Terry, the series of papers by EKPV are all devoted to schrodinger equation, since in the free case, the solution can be essentially expressed by the Fourier transform of the initial data, thanks to the Hardy uncertainty principle, we can obtain the quantitative uniqueness if we know the behavior of the solution at 2 different times. I'm interested to know if such idea can generalize to other types of evolution equation, and the solution may only be expressed by some convolution kernel which may be oscillatory, and still have quantitative uniqueness result. Do you think it is possible? $\endgroup$ – Tomas Jan 17 '14 at 4:47
  • $\begingroup$ Looking forward to go through the papers you suggested (...and thanks for understanding my intentions in the face of a confusing statement of the problem - I amended the question accordingly). $\endgroup$ – Maurizio Jan 17 '14 at 8:46

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