It is always expected that tunneling coupling splits the energy of the eigenstates in proportion to its magnitude. This is also true for bound states in the continuum (BICs) in the waveguide strongly coupled to a Fabry-Pérot resonator. However, in the case of weak coupling, giant enhancement of the splitting takes place. Within the tight-binding model, we show that, in the limit of a weak coupling, the BIC formation conditions take a universal form independent of the particular structure of the periodic perturbations. Moreover, BIC energies become almost uniformly distributed over the energy band of the waveguide in agreement with the Erdős-Turán theorem. Analytical conclusions are supported by numerical calculations of two-dimensional waveguides with finite periodic modulation for both quantum-mechanical and optical cases. The results present a new approach to BIC construction and may pave the way towards quantum design of large-scale integrated systems of giant atoms with spatially distributed coupling to waveguides.

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• Bound states in the continuum
• Quantum transport
• Quantum wires
• Waveguides
• Tight-binding model
• Transfer matrix method