How to prove that map $f\mapsto u$ from initial value to solution of Schrödinger equation is continuous map of $S(R^n)$ to $C^\infty(R^n,R)$? Thanks in advance.
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What is $S(R^n)$? What kinds of topology do you propose on these spaces? – Robert Lewis Apr 04 '14 at 21:49
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S is Schwartz space. – SOM Apr 04 '14 at 21:55
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I suspected a much. Thanks. – Robert Lewis Apr 04 '14 at 21:58
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I'm struggling with problems from Rauch book. :'( – SOM Apr 04 '14 at 22:03
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$$ \hat u(x,t) = \int_{\mathbb R^n} \exp(it|x-y|^2) \hat f(y) \, dy \\= \exp(i|x|^2) \int_{\mathbb R^n} \exp(-2it x\cdot y) \exp(it|y|^2) \hat u(y) \, dy \\= \exp(i|x|^2) \hat h(2tx) ,$$ where $h(x) = \exp(it|x|^2) \hat u(x)$. Now use the fact that the Fourier transform, and multiplying by $\exp(it|x|^2)$, both map $\mathcal S$ into $\mathcal S$.
Stephen Montgomery-Smith
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