Bound states in the continuum and long-lived electronic resonances in mesoscopic structures

A. bound state eigenfunction is defined here to be strictly localized within a subspace of the structure under study and has no decreasing behavior. Its eigenwavelength can be within state continua. Bound states in the continuum (BICs) and long-lived resonances have become a unique way to produce the extreme localization of electronic waves. We present a theoretical and numerical demonstration of semi-infinite bound states in the continuum (SIBICs) and long-lived resonances in a ringlike electronic microresonator coupled to a finite stub and to two electronic rib/ridge waveguides, together with their existence conditions. This structure is composed of a closed loop of length $L$, a finite stub of length $L_1$ and two semi-infinite leads. SIBICs localized in a semi-infinite subspace domain induce transmission zeros. Others induce transmission ones in the middle of long-lived resonances. The BICs correspond to localized resonances of infinite lifetime inside the structure, without any leakage into the surrounding leads. When BICs exist within state continua, they induce Fano resonances exhibiting sharp peaks in the transmission spectra and in the variation of the density of states for specific values of the stub length $L_1$. This enables one to regulate these resonances by means of this length. The obtained results take due account of the state number conservation between the final system and the reference one. This conservation rule enables one to find all the states of the final system and among them the bound in the continuum ones. The analytical results are obtained by means of the Green's function technique. The structures and the long-lived resonances presented in this paper may have potential applications due to their high sensitivities to weak perturbations, in particular in sensing, wave filtering, and microelectronic devices.