Barut-Girardello coherent states for sp(N,C) and multimode Schr\"odinger cat states

Overcomplete families of states of the type of Barut-Girardello coherent states (BG CS) are constructed for noncompact algebras $u(p,q)$ and $sp(N,C)$ in quadratic bosonic representation. The $sp(N,C)$ BG CS are obtained in the form of multimode ordinary Schr\"odinger cat states. A set of such macroscopic superpositions is pointed out which is overcomplete in the whole $N$ mode Hilbert space (while the associated $sp(N,C)$ representation is reducible). The multimode squared amplitude Schr\"odinger cat states are introduced as macroscopic superpositions of the obtained $sp(N,C)$ BG CS.}