I need to calculate the following integral (which is divergent):

\begin{equation} I(m,C)=\int_{-\infty}^\infty {\rm d}\omega\int_{\rm space}{\rm d^3 k}\ln\left(\omega-\frac{k^2}{2m}+C+i\epsilon\right) \end{equation} where $m$ and $C$ are constants, and $\epsilon$ is an infinitesimal positive number.

How should I regularize this integral to get $I$ as a meaningful function of $m$ and $C$?

  • $\begingroup$ If you take enough derivative with respect to $C$, the integral will be convergent. You can then integrate back, and fix the constant using limiting cases. $\endgroup$ – Adam Mar 28 '14 at 1:10
  • $\begingroup$ @Adam: After taking derivatives, what should the integral over $\omega$ give? $\endgroup$ – Mr. Gentleman Mar 28 '14 at 1:13
  • $\begingroup$ I think there should be a convergence factor like $e^{\omega 0^+}$ coming from the ordering of the field operators. Have a look at the Appendix G of R. B. Diener, et al., Phys. Rev. A, 77, 023626 (2008) (or find the arxiv free version). They do this kind of integral for a more complicated case, but you should be able to guess the result. $\endgroup$ – Adam Mar 28 '14 at 1:51

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