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What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks.

The operator I'm talking about is $$L=\partial_{tt}-\Delta_x$$ and for fundamental solution I mean a distribution $E$ (temperate) which satisfies $LE=\delta$ where $\delta$ is the Dirac distribution.

For $n=1$ one of such $E$ is $E(x,t)=(1/2)H(t)H(t^2-x^2)$ where $H$ is the Heaviside function.

I'm looking for expressions in higher dimensions.

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You need to give more details. If you don't reveal an explicit description of "the" wave operators (except that operators do not have solutions, fundamental or otherwise -- equations do), you can't expect to get any explicit formulas in response. –  Henning Makholm Jun 28 '12 at 15:39
    
Does $u=E$? Is delta some specific function or arbitrary or what? –  Matt Jun 28 '12 at 16:01
    
Have a look here for starts: math.ucsd.edu/~lindblad/110b/l17.pdf math.ucsd.edu/~lindblad/110b/l18.pdf –  night owl Jun 28 '12 at 16:29
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2 Answers 2

Higher dimensional cases will follow from Poisson's and Kirchhoff's formulas.

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What you are calling $E$ here, sounds like the Green's function of the wave operator....look at the bottom of the wiki page D'Alembert operator for the explicit form.

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The Wikipedia entry only applies for the case $n = 3$. –  Willie Wong Oct 3 '12 at 15:17
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