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Show that he distribution given by the locally integrable function $\dfrac{1}{2} e^{|x|}$ is a fundamental solution of the differential operator $\begin{equation} -\dfrac{\partial^{2}}{\partial x^{2}} + id \end{equation}$ on $\mathbb{R}^{1}$

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You can check directly by noting the fact that $$ \frac{d}{dx}e^{|x|}=(H(x)+H(-x))e^{|x|} $$ where H is the Heaviside function. On the other hand,you can use fourier transform to get the desired result,in fact,let u satisfies $$ (1-\frac{d}{dx^2})u=\delta $$ Take fourier transform on both sides,then get $$ \hat{u}=\frac{1}{1+x^2} $$ then the result follows easily.

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