14 imation). See also the discussion in Sec.II.D. tical Physics , edited by E. Tirapegui (Reidel, Dordrecht, 1981), pp. 237-245. [40] M. Miyamoto, Phys. Rev. A 72 , 063405 (2005). [41] For Im( E$_{a}$ ) = 0, we recover the Hermitian limit of the FFA Hamiltonian. In this case a divergence of the resol- vent for z → E 0 , obtained from Eq.(20) with V ( E 0 ) = 0, corresponds to a bound state embedded into the contin- uum (see, e.g., [39, 40]) rather than to a spectral singu- larity of H . [42] In fact, if z were a real-valued root of Eq.(18) outside the interval ( E 1 , E 2 ), according to Eq.(12) one would have Σ( z ) = Δ( z ) because V ( z ) = 0. Hence, Σ( z ) turns out to be real-valued. To satisfy Eq.(18), the condition Im( E$_{a}$ ) = 0 must be thus satisfied. This means that a real-valued root z of Eq.(18) necessarily requires H to be Hermitian. [43] Since the divergence of G ( z ) at z = E 0 on the continu- ous spectrum occurs when z approaches E 0 solely from one side (either from the top or from the bottom) of the complex plane, E 0 does not belong to the point spectrum of H , rather it is a spectral singularity. In fact, if E 0 were an eigenvalue of H corresponding to a square-integrable eigenfunction |E 0 〉 , denoting by |E † 0 〉 the eigenfunction of the adjoint H † corresponding to the same eigenvalue and taking in Eq.(5) | χ 〉 = |E † 0 〉 and | ϕ 〉 = |E 0 〉 , one would obtain G $_{χ,ϕ}$( z ) = 〈E † 0 |E 0 〉 / ( z − E 0 ). Since 〈E † 0 |E 0 〉 is finite, it would then follow that G $_{χ,ϕ}$( z ) should diverge when both z = E 0 + i 0 + and z = E 0 − i 0 $^{+}$. [44] For a featureless continuum, for which Δ( E ) and V ( E ) are smooth functions of E , the resonance curve (39) has a Lorentzian shape and the quantum system, intially prepared in state | a 〉 , would decay into the continuum following a (nearly) exponential decay law with lifetime 1 / [ πV ( E$_{a}$ )] (Weisskopf-Wigner or Breit-Wigner approx- [51] R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32 , 2632 (2007); K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, Phys. Rev. Lett. 100 , 103904 (2008); A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103 , 093902 (2009). [50] At the coalescent point, the resolvent G ( z ) has a second- order pole at z = 0. [49] More precisely, far from the lattice boundary a wave packet of the form c$_{n}$ ( t ) = ∫ dkQ ( k ) exp[ − ikn − iE ( k ) t ], with spectrum Q ( k ) narrow at around k = k 0 (0 < k 0 < π ), propagates along the lattice with a group ve- locity v$_{g}$ = dn/dt = − 2 κ 0 sin k 0 < 0, i.e. the wave packet is incident onto the lattice boundary. Conversely, a wave packet formed by the superposition c$_{n}$ ( t ) = ∫ dkQ ( k ) exp[ ikn − iE ( k ) t ], with spectrum Q ( k ) narrow at around k = k 0 , propagates along the lattice with a group velocity v$_{g}$ = 2 κ 0 sin k 0 > 0, i.e. it is reflected from the lattice edge. [48] O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103 , 030402 (2009). Im Σ( z = E ± i 0 $^{+}$) = ∓ πV ( E ), according to Eq.(12). [47] F. Dˇ ogan, W. Kim, C.M. Blois, and F. Marsiglio, Phys. Rev. B 77 , 195107 (2008). [45] We assume here that the spectrum of H is purely con- tinuous, i.e. we assume that there are not bound states (outside or embedded in the continuum). In this case, for the Hermitian FFA model the decay of the survival probability P ( t ) is always complete. [46] The proper determination of the square root on the right hand side of Eq.(50) must be chosen such that