Negative eigenvalues of the Y-type chain of weakly coupled ball resonators

Spectral properties of a system are strongly associated with its geometry. The spectral problem for the Y-bent chain of weakly-coupled ball resonators is investigated. The Y-bent system can be described as a central ball linking three chains consisting of balls of the same radius. There is a $\delta$-coupling condition with parameter $\alpha$ at every contact point. Specifically, it is assumed that the axis passing through the center of each ball lies in the same plane and the centers of balls that are the closest to the central ball form an equilateral triangle. The transfer-matrix approach and the theory of extensions are employed to solve the spectral problem for this system. It is shown that such system with a certain value of parameter $\alpha$ has at most one negative eigenvalue in the case of $\delta$-coupling in contact points.