We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard which is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.

Using formalism of effective Hamiltonian we consider bound states in continuum (BIC). They are those eigen states of non-hermitian effective Hamiltonian which have real eigen values. It is shown that BICs are orthogonal to open channels of the leads, i.e. disconnected from the continuum. As a result BICs can be superposed to transport solution with arbitrary coefficient and exist in propagation band. The one-dimensional Aharonov-Bohm rings that are opened by attaching single-channel leads to them allow exact consideration of BICs. BICs occur at discrete values of energy and magnetic flux however it's realization strongly depend on a way to the BIC's point.Comment: 5 pgaes, 4 figure

The physics of nanoscale systems has advanced rapidly over the last few years. A consistent description of these small systems is a challenging task for quantum theory since their properties may be influenced strongly by attaching leads to them [1][2][3][4][5][6][7][8][9]. They are simulated often by means of quantum billiards. When the cavity is not fully opened, the propagation of the mode is restricted to energies at which the overlap integral between the wave functions of the resonance states and the channel modes has a non-vanishing value. In the case of well isolated resonances, the electron can propagate therefore only at the energies of the resonance states ("resonance tunneling"). Due to the coupling between the internal states of the cavity and the channel mode, the states get widths. When the coupling is sufficiently strong, the resonances start to overlap and to interact via the channel mode.As a consequence, some redistribution in the resonance states of the cavity takes place. It 1