Interfering resonances and bound states in the continuum

We use Feshbach's theory of resonances to demonstrate that bound states in the continuum (BIC's) can occur due to the interference of resonances belonging to different channels. If two resonances pass each other as a function of a continuous parameter, then interference causes an avoided crossing of the resonance positions and for a given value of the continuous parameter one resonance has exactly vanishing width and hence becomes a BIC. The condition for a BIC relates the positions of the noninterfering resonances with the coupling matrix elements between the various channels. In the neighborhood of the BIC point one resonance remains anomalously narrow for a finite range of values of the separation of the noninterfering resonances. Whether or not two resonances interfere is not directly related to whether or not they overlap. All these results, including the occurrence of exactly bound states in the continuum, are not consequences of approximations inherent in Feshbach's theory but are general features of a coupled-channel Schr\"odinger equation with only one open channel. We illustrate the results in a simple but realistic model, where all matrix elements involved can be calculated analytically. We also discuss the case of coupled Coulombic channels where BIC's are caused by perturbations interfering with a Rydberg series of autoionizing resonances. Below the continuum threshold the analogy to a BIC is an infinitely narrow perturbation of the bound-state spectrum. Near such an infinitely narrow perturbation we may observe approximate level crossings.