de Branges-Rovnyak spaces which are complete Nevanlinna-Pick spaces

We consider de Branges-Rovnyak spaces of a considerably large class of reproducing kernel Hilbert spaces and find a characterization for them to be complete Nevanlinna-Pick spaces. This extends as well as recovers earlier characterizations obtained for the Hardy space over the unit disc (\cite{Chu}) as well as for the Drury-Arveson space over the unit ball (\cite{Jesse}). Our characterization takes a complete form for the particular cases of the Hardy space over the polydisc and the Bergman space over the disc. We show that a non-trivial de Branges-Rovnyak space, associated to a contractive multiplier, of the Hardy space over the bidisc or the Bergman space over the unit disc is a complete Nevanlinna-Pick space if and only if it is isometrically isomorphic to the Hardy space over the unit disc. On the contrary, it is shown that non-trivial de Branges-Rovnyak spaces of the Hardy space over the $n$-disc with $n\ge 3$ are never complete Nevanlinna-Pick spaces.