Question regarding a special identity for $2\pi\delta(E-\epsilon_\alpha)$

Question

I am reading Datta's book about Quantum Transport at the moment and I stumbled over an identity for the Dirac delta distribution, which is correct since it fullfils all the requirements for the Dirac delta distribution as discussed here, but looks so arbitrary to me that I was wondering how this special identity came up.

Datta writes

In evaluating the spectral function it is convenient to make use of the identity $$2\pi\delta(E-\epsilon_\alpha)= \left[ \frac{2\eta}{(E-\epsilon_\alpha)^2+\eta^2}\right]_{\eta\to0^+} = \quad...$$ [...]

He does this to connect the spectral function with Green's function. He also mentions this step as if it is a rather common identity used in quantum transport physics, however after looking through some lists of common identities I couldn't find anything that resembles the structure of it.

My question now is, if there is an underlying concept for coming up with that identity or if it just proved itself as useful in that particular case of connecting the spectral function with Green's function after trial&error-ring a bunch of different approaches.

Answer 1

This is one of the possible representations of the delta-function. One also often uses its another form: $$ \frac{1}{\omega \pm i\eta} = \frac{\omega}{\omega^2+\eta^2} \mp \frac{i\eta}{\omega^2+\eta^2}= \mathcal{P}\frac{1}{\omega} \mp i\pi\delta(\omega) $$ where $\mathcal{P}$ means the principal value of the integral where the equation appears.

These are mathematical identities widely used beyond the quantum transport or even the semiconductor physics, so it is unlikely to be listed as special to either of these fields.

Mathematically, delta function is a generalized function, defined as a limit of a series of functions, satisfying sertain conditions (peaked, unit integral area, etc.) Thus, different series of functions converge to the same limit (just like different series of numbers may converge to the same number). Another frequent choice is a gaussian: $$ \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2\sigma^2}} \longrightarrow_{\sigma\longrightarrow 0^+} \delta(x) $$

Answer 2

This function is what is known as a nascent $\delta$ function. You can read more about these in the wiki article, https://en.wikipedia.org/wiki/Dirac_delta_function#nascent_delta_function

The idea is that a series of smooth functions may converge to the $\delta$ distribution, giving you an idea of how to respresent the $\delta$ distribution on smooth function spaces.

So what you have is a series of functions $\delta_{\eta}$ depending on some paramter $\eta$ that basically provide $$\int_{U} \textrm{d}x\, f(x)\delta_{\eta}(x) \to \begin{cases}f(0), \qquad & 0\in U\\0, \qquad &\textrm{otherwise} \end{cases}$$ as $\eta \to 0$ (sometimes $\eta \to \infty$) for all continuous funtions $f(x)$ and all open sets $U$.

There are a set of requirements that a nascent $\delta$ function must meet. You can check all of these for the series $$\delta_{\eta}(x) = \frac{1}{\pi} \frac{\eta}{x^2 + \eta^2}.$$ They are

1. Positivity: $\delta_{\eta}(x) \ge 0$ is easy
2. Unit integral: $\int \textrm{d}x\, \delta_{\eta}(x) = 1$ can be calculated by using $$(\tanh^{-1}(x))' = \frac{1}{x^2 + 1},$$ so $$\int \textrm{d}x\, \frac{\eta}{x^2 + \eta^2} = \int \frac{\textrm{d}x}{\eta}\, \frac{1}{(x/\eta)^2 + 1} = [\tanh^{-1}(x)]_{-\infty}^\infty = \pi$$
3. Vanishing integral outside of key domain: For all $\epsilon > 0$ the integral $$\int_{\mathbb{R}\backslash (-\epsilon,\epsilon)} \textrm{d}x\, \delta_{\eta}(x) \to 0,$$ which follows from $\tanh^{-1}(x) \to 0$ for $x \to 0$. This condition ensures that there is only one relevant peak centered at $0$ that grows narrower with decreasing $\eta$.