# Gradiant and exponent of trace of matrices

1. Dec 28, 2011

### timb00

hi everybody,

I'm currently work on a simple problem where i want to show that
$$\int d[H] ~ \exp\{-1/(2v^2)tr(H^2)\} \exp\{i tr(HK)\} = \exp\{-1/(2v^2)tr(\bigtriangledown_K^2)\} \int d[H] ~\exp\{i tr(HK)\}$$

where $d[H]$ is the lebesgue measure on the space of NxN matrices. This is easy and was no problem when one use simples rules form fourier analysis. Now I would like to replace K, since it is hermitian, by a product of a Nx2k matrix $A$ and its hermitian conjugated $A^{\dagger}$. May anyone of you has an idea how to express the integral above with the aid of differential operators depending on the components of $A$ and $A^{\dagger}$. Or maybe could someone of you tell me, what is the problem in finding such an expression from mathematical point of view.

best Timb00