Manifesting the evolution of eigenstates from quantum billiards to singular billiards in the strongly coupled limit with a truncated basis by using RLC networks.

The coupling interaction between the driving source and the RLC network is explored and characterized as the effective impedance. The mathematical form of the derived effective impedance is verified to be identical to the meromorphic function of the singular billiards with a truncated basis. By using the derived impedance function, the resonant modes of the RLC network can be divided into the open-circuit and short-circuit states to manifest the evolution of eigenvalues and eigenstates from closed quantum billiards to the singular billiards with a truncated basis in the strongly coupled limit. The substantial differences of the wave patterns between the uncoupled and strongly coupled eigenmodes in the two-dimensional wave systems can be clearly revealed with the RLC network. Finally, the short-circuit resonant states are exploited to confirm that the experimental Chladni nodal-line patterns in the vibrating plate are the resonant modes subject to the strong coupling between the oscillation system and the driving source.