An almost Complex Structure with Norden Metric on the Phase Space

On the total space of the cotangent bundle of a Riemannian manifold, we construct a semi-Riemannian metric $G$, with respect to which an almost complex structure $J$ introduced by Oproiu and Poro\cb{s}niuc is self-adjoint. The structure $(J,G)$ turnes out to be an almost complex structure with Norden metric (this notion is known in the literature from Norden's papers). The semi-Riemannian context is different from the Riemannian one, as it is pointed out by Duggal and Bejancu in their monograph. We study this structure and provide some necessary and sufficient conditions for it to be a K\"ahler structure with Norden metric.