Suppression of Feshbach Resonance Widths in Two-Dimensional Waveguides and Quantum Dots: A Lower Bound for the Number of Bound States in the Continuum

The possibility of having bound states in the continuum was proposed by von Neumann and Wigner shortly after the birth of quantum mechanics. However, it is still considered a rare special phenomenon. Here we show how a lower bound to the number of bound states in the continuum can be calculated as a function of the open two-dimensional potential parameters of quantum dots and optical waveguides. The proof we present here holds for potentials that are symmetric in the perpendicular direction to the exits and entrances of the quantum dots or the waveguides.