Unified theory of resonances and bound states in the continuum in Hermitian tight-binding models

We study the transport properties of an arbitrary two-terminal Hermitian system within a tight-binding approximation and derive an expression for the transparency in a form that enables one to determine the exact energies of the perfect (unity) transmittance, zero transmittance (Fano resonance), and bound state in the continuum (BIC). These energies correspond to the real roots of two energy-dependent functions that are obtained from two non-Hermitian Hamiltonians: the Feshbach effective Hamiltonian and the auxiliary Hamiltonian, which can be easily deduced from the effective one. BICs and scattering states are deeply interconnected. We show that the transformation of a scattering state into a BIC can be formally described as a “phase transition” with a divergent generalized response function. Design rules for quantum conductors and waveguides are presented. These rules describe the structures exhibiting coalescence of both resonances and antiresonances resulting in the formation of almost rectangular transparency and reflection windows. The results can find applications in the construction of molecular conductors, broad-band filters, quantum heat engines, and waveguides with controllable BIC formation.