We study transport properties of an arbitrary two terminal Hermitian system within a tight-binding approximation and derive the expression for the transparency in the form, which enables one to determine exact energies of 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's effective Hamiltonian and the auxiliary Hamiltonian, which can be easily deduced from the effective one. BICs and scattering states are deeply connected to each other. We show that transformation of a scattering state into a BIC can be formally described as a "phase transition" with divergent generalized response function. Design rules for quantum conductors and waveguides are presented, which determine 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 construction of molecular conductors, broad band filters, quantum heat engines and waveguides with controllable BIC formation.
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We study the reflectivity of Casimir-van der Waals potentials, which behave as ϪC 4 /r 4 at large distances and as ϪC 3 /r 3 at small distances. The overall behavior of the reflection amplitude R depends crucially on the parameter ϭͱ2MC 3 /(បͱC 4 ) which determines the relative importance of the Ϫ1/r 3 and the Ϫ1/r 4 parts of the potential. Near threshold, Eϭប 2 k 2 /(2M )→0, the reflectivity is given by ͉R͉ϳexp(Ϫ2bk), with b depending on and the shape of the potential at intermediate distances. In the limit of large energies, ln͉R͉ is proportional to Ϫk 1/3 with a known constant of proportionality depending only on C 3 . For small values of , the reflectivity behaves as for a homogeneous Ϫ1/r 3 potential in the whole range of energies and does not depend on C 4 or the shape of the potential beyond the Ϫ1/r 3 region. For moderate and large values of , the reflectivity depends on C 4 and on the potential shape. For sufficiently large values of , which are ubiquitous in realistic systems, there is a range of energies beyond the near-threshold region, where the reflectivity shows the high-energy behavior appropriate for a homogeneous Ϫ1/r 4 potential, i.e., ln͉R͉ is proportional to Ϫͱk with a proportionality constant depending only on C 4 . This conspicuous and model-independent signature of the Casimir effect is illustrated for the reflectivities of neon atoms scattered off a silicon surface, which were recently measured by Shimizu ͓Phys. Rev. Lett. 86, 987 ͑2001͔͒.
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