# Fundamental solution of a particular differential operator

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