Timeline for eigenfunction of heat operator.

8 events

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May 29, 2012 at 6:48	comment	added	spr		I am not talking about $L^2$-eigenfunctions. There are $L^\infty$-eigenfunctions of $\Delta$ on $\mathbb R^n$: $x↦e^{−i(x_1y_1+⋯+x_ny_n)}$. There are eigenfunctions in some other $L^p$ spaces depending on $n$. Any eigenfunction of $\Delta$ on $\mathbb R^n$ is clearly an eigenfunction of $e^{-t\Delta}$. I am asking if there is an eigenfunction of $e^{-t\Delta}$ on $\mathbb R^n$ for a fixed $t>0$ which is not an eigenfunction of $\Delta$.
May 29, 2012 at 5:25	history	edited	Denis Serre	CC BY-SA 3.0	added 86 characters in body
May 28, 2012 at 15:13	history	edited	Denis Serre	CC BY-SA 3.0	added 25 characters in body
May 28, 2012 at 15:12	comment	added	Denis Serre		I apologize for the typos and I edit ...
May 28, 2012 at 13:38	comment	added	Emilio Pisanty		(terrific answer, otherwise!)
May 28, 2012 at 13:38	comment	added	Emilio Pisanty		Shouldn't that sum go from $m=1$ and add up to $-\textrm{ln}(\lambda)$? The eigenvectors of $\Delta$ and $e^{t\Delta}$ should hopefully match, but not necessarily so the corresponding eigenvalues.
May 28, 2012 at 13:07	comment	added	Jon		Do you mean $\Delta$ rather than $\delta$ in the second paragraph?
May 28, 2012 at 12:51	history	answered	Denis Serre	CC BY-SA 3.0