Heaviside function
Properties

Functional analysis MOC

Heaviside function

The Heaviside function is defined as fun

𝐻 : ℝ \to ℝ 𝑥 \mapsto ⎧ { {⎨ { {⎩ 0 𝑥 < 0 1 𝑥 > 0 12 𝑥 = 0

Properties

$𝐷 𝐻 = 𝛿$ where $𝛿$ is the Dirac delta, hence $𝐻$ is a Green’s function of $𝐷$ .

Proof of 1

Let $𝜌 (𝑥)$ be an arbitrary non-pathological function
$\int \infty - \infty 𝑑 𝑑 𝑥 𝐻 (𝑥 - 𝑥') 𝜌 (𝑥') 𝑑 𝑥' = 𝑑 𝑑 𝑥 \int \infty - \infty 𝐻 (𝑥 - 𝑥') 𝜌 (𝑥') 𝑑 𝑥' = 𝑑 𝑑 𝑥 [\int 𝑥 - \infty 𝐻 (𝑥 - 𝑥') 𝜌 (𝑥') 𝑑 𝑥' + \int \infty 𝑥 𝐻 (𝑥 - 𝑥') 𝜌 (𝑥') 𝑑 𝑥'] = 𝑑 𝑑 𝑥 \int 𝑥 - \infty 𝜌 (𝑥') 𝑑 𝑥' = 𝜌 (𝑥) = \int \infty - \infty 𝛿 (𝑥 - 𝑥') 𝜌 (𝑥') 𝑑 𝑥$
as claimed.

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Green's function
QM of a particle in a 1D Dirac delta potential

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