Bound states in the continuum (BICs) has emerged as a significant research focus in electronics due to its exceptionally high quality factor (Q-factor). BICs (known also as trapped modes) are not observable from the spectrum due to their non-radiative property. However, they can exist only under a specific choice of the materials or geometrical parameters of the structure. In this paper a BIC eigenfunction is defined to be strictly localized within a subspace of the cavity structure under study and has no leakage behaviour. Its eigen wavelength can be within state continua. BICs and long-lived resonances (LLR) have become a unique way to produce the extreme confinement of electronic waves. We present a theoretical and numerical demonstration of semi-infinite bound states in the continuum (SIBICs) and LLR in a two ring-like electronic micro-cavity coupled to two electronic rib/ridge wave-guides, together with their existence conditions. This structure is composed of two tangent closed loops of lengths \(L_1\) and \(L_2\) , and two semi-infinite leads. SIBICs are localized in a semi-infinite subspace domain induced transmission zeros. Other induce transmission ones in the middle of long-lived resonances. The BICs correspond to localized resonances of infinite lifetime inside the cavity, without any leakage into the surrounding leads. When BICs exist within state continua, they induce Fano resonances exhibiting sharp peaks in the transmittance spectra and in the variation of the density of states (VADOS) for specific values of the geometrical parameters \(L_1\) and \(L_2\) . We demonstrate that the condition for the existence of the BICs is to make the lengths \(L_1\) and \(L_2\) commensurate with each other. This enables to control the resonances by engineering these lengths. Finally, such a two-tangent loops cavity can be designed to realize near-perfect absorption for some frequencies. The results obtained take due account of the state number conservation between the final system and the reference one. This conservation rule enables to find all the states of the final system and among them the BIC ones. The analytical results are obtained by means of the Green’s function technique. The cavity structure and the LLR presented in this work may have potential applications due to their high sensitivities to weak perturbations, in particular in sensing and wave filtering.
