  We have theoretically investigated the time-symmetry breaking phase
transition process for two discrete states coupled with a one-dimensional
continuum by solving the nonlinear eigenvalue prob- lem for the effective
Hamiltonian associated with the discrete spectrum. We obtain the effective
Hamiltonian with use of the Feshbach-Brillouin-Wigner projection method. Strong
energy depen- dence of the self-energy appearing in the effective Hamiltonian
plays a key role in the time-symmetry breaking phase transition: as a result of
competition in the decay process between the Van Hove singularity and the Fano
resonance, the phase transition becomes a higher-order transition when both the
two discrete states are located near the continuum threshold.
