Bound states in the continuum in open spherical resonator

We consider the bound states in the continuum (BICs) or embedded trapped modes in an open spherical acoustic resonator. The eigenfrequencies of closed resonator are2l+1<i-fold degenerated, wherelis the orbital index. An attachment of two cylindrical waveguides lifts this degeneracy and transforms the eigenfrequencies into resonances whose real parts depend on the position of the waveguides. When the waveguides are angled by pi<i, variation over that angle gives rise to avoided crossings of resonant modes with differentlto result in the Friedrich-Wintgen BICs. For = pi there might be only the symmetry protected BICs. When three waveguides are connected to the spherical resonator the Friedrich-Wintgen BICs occur due to the avoided crossings of resonant modes with the samelbut different azimuthal indices-l <= m <= l.