Dear all,(adsbygoogle = window.adsbygoogle || []).push({});

In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modelled as a mass connected to neighbouring atoms through springs). He gives the green function as

## G = \frac{2}{\omega^2 + \sqrt{\omega^4 + 4 k \omega^2}} ##

From this the intention is to find the spectral density of states. He directly gives it as

## \rho = \frac{1}{2 \pi} \frac{\sqrt{\omega^4 + 4 k \omega^2}}{2 k \omega^2} ##.

I have not been able to show this.

Earlier in the chapter he uses the standard representation for the spectral density of states as

## G - G^* = 2 \pi \rho ##.

Using this I attempted the following way. Let ## \omega^2 = z ## (he had used such a representation earlier, hence I tried this), which gives:

## G(z) = \frac{2}{z + \sqrt{z^2 + 4 k z}} ##

which can also be written as:

## G(z) = \frac{ z - \sqrt{z^2 + 4 k z}}{- 2 k z} ##

Because of the ## \sqrt{z}## factor there is a branch cut along negative real axis. That's probably the only reason why ## G - G^*## would have a non-negative value - since there would be a non-zero difference in the values across the branch cut.

So what we need is :

## \lim_{\delta \rightarrow 0} G(z+ i \delta) - G(z - i \delta) ##

This gives:

## \rho = \lim_{\delta \rightarrow 0} \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k}{z+ i \delta}} - 1}{ 2 k } - \frac{ \sqrt{1 + \frac{4 k }{z - i \delta}} - 1}{ 2 k } \right) = \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k }{z+ i \delta}} - \sqrt{1 + \frac{4 k }{z - i \delta}}}{ 2 k } \right) ##

If you take the limit, this just goes to zero.

What am I doing wrong?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Green's function and density of states

**Physics Forums | Science Articles, Homework Help, Discussion**