Inverse problem in femtoscopic correlation functions: The $T_{cc}(3875)^+$ state

We study here the inverse problem of starting from the femtoscopic correlation functions of related channels and analyze them with an efficient tool to extract the maximum information possible on the interaction of the components of these channels, and the existence of possible bound states tied to this interaction. The method is flexible enough to accommodate non-molecular components and the effect of missing channels relevant for the interaction. We apply the method to realistic correlation functions for the $D^{*+}D^0$ and $D^{*0}D^+$ channels derived consistently from the properties of the $T_{cc}(3875)^+$ and find that we can extract the existence of a bound state, its nature as a molecular state of the $D^{*+}D^0$ and $D^{*0}D^+$ channels, the probabilities of each channel, as well as scattering lengths and effective ranges for the channels, together with the size of the source function, all of them with a relatively good precision.