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Let us consider the linear Schrödinger equation in $\mathbb{R}^N$

$$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$

with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ be its solution.

We have that

$$\frac{1}{T}\int_0^Te^{it\Delta}fdt\rightarrow 0\quad\mbox{in }L^2(\mathbb{R}^N)$$

The proof I have is "geometric", based on the following idea:

The operator (once one proved it exists)

$$P:\,f\mapsto\lim_{t\rightarrow +\infty}\frac{1}{T}\int_0^Te^{it\Delta}fdt$$

satisfies $P\circ e^{it\Delta}=P$ and $e^{it\Delta}\circ P=P\quad \forall t\in\mathbb{R}.$

This implies $\mathrm{Im}\, P = \mathrm{Fix}\{e^{it\Delta}\}_{t\geq 0} = \mathrm{Ker}\, i\Delta=\{0\}$, that is $P=0$.

I'm looking for a more analytical proof, based on the decay estimates for ${e^{it\Delta}}$. Infact I would try to understand under which hypotesis I can extend the result to the nonlinear case, where a geometric approach seems to be unavailable. Does anyone has some ideas?

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  • $\begingroup$ For a proof use fourier transform or resolution of identity of normal operators. However, I don't know if this is useful for the nonlinear case. $\endgroup$
    – jjcale
    Aug 19, 2013 at 19:16
  • $\begingroup$ I can't figure it out. Can you show me a proof that use fourier transform? $\endgroup$
    – hispac
    Aug 22, 2013 at 0:20
  • $\begingroup$ Your operator acts on the fourier transform $\hat{f}(k)$ as a multiplication by $(e^{-i k^{2}T}-1)/(-i T k^{2})$ $\endgroup$
    – jjcale
    Aug 22, 2013 at 21:02

1 Answer 1

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Ruess and Summers have a generalization of the geometric ideas you describe to nonlinear contraction semigroups. I suggest you take a look.

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  • $\begingroup$ I took a look to the work of Ruess and Summers. It's very interesting. However i'm searching for different tools (especially harmonic analysis) tto attack the problem. Do you have any sugguestions? $\endgroup$
    – hispac
    Aug 22, 2013 at 0:21

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