Abstract
This paper is concentrated on the frame potential of finite frames in \(n\) -dimensional Hilbert space \(\mathcal{H}_{n}\) . More precisely, We define the relative frame potential of two sequences \(\mathcal{F}\) and \(\mathcal{G}\) in \(\mathcal{H}_{n}\) , which is denoted by \(\widetilde{FP}\left(\mathcal{F},\mathcal{G}\right)\) . For dual frames \(\mathcal{F}\) and \(\mathcal{G}\) in \(\mathcal{H}_{n}\) , we show that \(\mathcal{G}\) is canonical dual frame of \(\mathcal{F}\) if and only \(\widetilde{FP}\left(\mathcal{F},\mathcal{G}\right)=n\) . A lower and upper bound for \(\widetilde{FP}\left(\mathcal{F},\mathcal{G}\right)\) is given, for the case that \(\mathcal{F}\) and \(\mathcal{G}\) are frames for \(\mathcal{H}_{n}\) . Also, using the relative frame potential of the sub sequences of a given frame, some equivalent conditions for its phase retrievability and its norm retrievability, are presented.
