Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

(propose a free ad)

I have a question regarding calculating the following integral with cutoff. $$ \int_{-\infty}^{\infty} \frac{d\omega}{|\omega|} \cos(\omega(\tau_i-\tau_j)-1)$$

How should I set up the correct cutoff so that the result of the integral becomes: $$-2\ln \frac{\tau_i-\tau_j}{\tau_c}$$ where $\tau_c$ is the short time cutoff~$E_F^{-1}$.

This calculation is from certain resonant tunneling problem of Luttinger liquid, for which you can obtain renormalization group flow equation.

rewrite the cosine in terms of exponenials, and add a tiny real part to the exponents.

In more detail, rewrite the integral as twice the integral from 0 to $\infty$, abbreviate $\tau_i-\tau_j$ by $t$, differentiate with respect to $t$ to get rid of the denominator, use $cos x = (e^{ix}+e^{-ix})/2$, and change $e^{iz}$ to $e^{iz-\epsilon\omega}$ to be able to perform the integration. Then set $\epsilon=0$ and integrate the result from $t=\tau_c \approx 0$ to $t=\tau_i$-$\tau_j$.

Could you explain a little bit more in detail?

I added some details.

So how do you handle the 0 for integral of (1/t), wouldn't that produce -infinity there?

In the final integration w.r. to $t$, one needs the cutoff mentioned.

user contributions licensed under cc by-sa 3.0 with attribution required