Observability of a string-beams network with many beams

We prove the direct and inverse observability inequality for a network connecting one string with infinitely many beams, at a common point, in the case where the lengths of the beams are all equal. The observation is at the exterior node of the string and at the exterior nodes of all the beams except one. The proof is based on a careful analysis of the asymptotic behavior of the underlying eigenvalues and eigenfunctions, and on the use of a Ingham type theorem with weakened gap condition [C. Baiocchi, V. Komornik and P. Loreti, Acta Math. Hung. 97 (2002) 55-95.]. On the one hand, the proof of the crucial gap condition already observed in the case where there is only one beam [K. Ammari, M. Jellouli and M. Mehrenberger, Networks Heterogeneous Media 4 (2009) 2009.] is new and based on elementary monotonicity arguments. On the other hand, we are able to handle both the complication arising with the appearance of eigenvalues with unbounded multiplicity, due to the many beams case, and the terms coming from the weakened gap condition, arising when at least 2 beams are present.