Multidimensional Gaussian-Fresnel integral related to a matrix inverse

Question

How would you evaluate the following multidimensional Gaussian-Fresnel integral? $$ I_{k,N;\lambda,\epsilon}(\mathbf{H})=\int_{\mathbb{R}^N} dx_1\cdots dx_N\ x_k^2\exp\left[-\frac{\mathrm{i}}{2}\sum_{i,j=1}^N x_i ((\lambda-\mathrm{i}\epsilon)\mathbf{1}-\mathbf{H})_{ij}x_j\right]\ , $$ for $H$ a real (invertible) symmetric matrix, $\lambda\in\mathbb{R}$ and $\epsilon>0$? It must be related somehow to $[\mathbf{H}^{-1}]_{kk}$, judging from a few references (e.g. eq. (10) in http://arxiv.org/pdf/1005.3712v3.pdf), but I have so far somehow failed to find a very convincing (extended) proof. If you have a good and neat reference, that would als be much appreciated. Thanks, folks.