On the Existence of Bound States in the Continuum Related to Destructive Interference

A bound state in the continuum (BIC) is an eigenmode of a linear operator with the eigenvalue in the continuous spectrum, i.e. the eigenvalue is an embedded eigenvalue. I recent years, there has been a significant research interest on BICs of electromagnetic waves in the photonics community, mainly due to their important applications in lasing, sensing, switching, filtering, nonlinear optics, etc. Theoretical works on BICs are concerned with the physical mechanisms leading to the formation of BICs, resonant and leaky states surrounding the BICs, the effect of structural perturbations on the BICs, etc. Except for simple BICs, such as those related to symmetry mismatch and separability, precise conditions on the existence of many BICs are not available. In particular, destructive interference was identified as the physical mechanism for many BICs, but the precise conditions that guarantee the total elimination of outgoing waves are typically unknown. In this project, we consider a few general models and try to identify conditions (sufficient conditions) under which the existence of BICs can be rigorously proved. Our work contributes to the theoretical foundation of the interdisciplinary research on BICs, and also provides useful guidance for realizing their practical applications.