11 IV. CONCLUSIONS In this work a comprehensive analysis of the spectral properties of a non-Hermitian extension of the Friedrichs- Fano-Anderson model has been presented. The FFA model generally describes the decay of a discrete state | a 〉 of energy E$_{a}$ coupled to a continuum of states. Here we have extended the ordinary model by allowing the energy E$_{a}$ to become complex-valued, with either Im( E$_{a}$ ) > 0 (the ’amplifying’ case) or Im( E$_{a}$ ) < 0 (the ’absorbing’ case). Contrary to the Hermitian FFA, it has been shown by a direct diagonalization procedure and by the analy- sis of the resolvent operator that spectral singularities in the continuous spectrum may exist for both the am- plifying and the absorbing non-Hermitian FFA model. The physical implications and relevance of spectral singu- larities have been discussed, in particular with reference to a tight-binding realization of the non-Hermitian FFA model that describes single-particle electronic or pho- tonic transport in a semi-infinite lattice attached to an impurity site with a complex energy. Different behaviors have been found for an amplifying and for an absorb- ing impurity site, reflecting the circumstance that the divergence of the resolvent G ( z ) appears when the spec- tral singularity is approached either from above or from below of the complex energy plane. For an amplifying impurity site, the spectral singularity manifests itself as a divergence of the reflection probability from the lattice boundary, a result which is the ’discrete’ analogous of the general result recently established by Mostafazadeh for wave scattering by complex potentials in the framework of the continuous Schr¨ odinger equation [11]. As com- pared to Ref.[11], here we have also clarified the phys- ical relevance of spectral singularities in the temporal domain, showing that in the amplifying non-Hermitian FFA Hamiltonian the spectral singularity of the resol- vent prevents the decay of state | a 〉 into the continuum, in spite of the absence of bound states. For an absorbing impurity site, we have shown that a spectral singularity corresponds to a zero of the reflection probability from the lattice boundary. This result is clearly not observable in the problem of wave scattering from complex barriers addressed in Refs. [11, 12], where the double-degeneracy of energy levels plays a major role and a spectral singu- larity can never correspond to the total absorption of the incident wave. Owing to the importance of the FFA model in different areas of physics, it is envisaged that the present anal- ysis may stimulate further theoretical and experimental studies aimed to investigate the unique features of non- Hermitian physical systems. In particular, recent results obtained in photonic systems based on coupled waveg- uides or arrays of coupled waveguides with controlled re- gions of optical gain and/or loss [51], indicate that en- gineered photonic systems might provide an accessible laboratory tool to experimentally observe spectral singu- larities. eral analysis presented in Sec.II.D, the survival probabil- ity P ( t ) = | c$_{a}$ ( t ) | 2 decays to zero in the absorbing case (Fig.9), and the existence of a spectral singularity does not basically influence the decay dynamics of state | a 〉 . Conversely, for an amplifying impurity site the occupa- tion probability P ( t ) does not decay to zero. Note that for κ$_{a}$/κ$_{0}$ < √$_{2 [like in Fig.8(a)],}$ P ( t ) converges to a steady state value, whereas for κ$_{a}$/κ$_{0}$ = √$_{2 the proba-}$ bility P ( t ) is an oscillating function [see Fig.8(b)]. The different behavior in the two cases is explained by observ- ing that, for κ$_{a}$/κ$_{0}$ = √$_{2 there is one spectral singularity}$ and thus in Eq.(45) there is only one pole that contributes to the non-decaying part of c$_{a}$ ( t ). On the other hand, for the somehow special case κ$_{a}$/κ$_{0}$ = √$_{2 the Hamiltonian}$$_{H}$ has two spectral singularities [see Eq.(57)], and in Eq.(45) there are two poles that contribute to the non-decaying part of c$_{a}$ ( t ). The interference of these two non-decaying terms explains the oscillatory behavior of P ( t ) in Fig.8(b) (see also [35]). FIG. 9: (color online) Same as Fig.8, but for an absorbing impurity site (Im( E$_{a}$ ) = − 1; other parameters as in Fig.8) . (b) (a) t occupationGLYPH<.notdef>probability P t ( ) 0 1 2 3 4 5 6 0.5 1 0 1 2 3 4 5 6 0.5 1 (b) (a) 0 5 10 15 20 1 2 3 4 5 0 5 10 15 20 0.5 1 1.5 2 t occupationGLYPH<.notdef>probability P t ( ) FIG. 8: (color online) Evolution of occupation probability P ( t ) = | c$_{a}$ ( t ) | 2 in a semi-infinite lattice with an amplifying impurity site for Re( E$_{a}$ ) = 0, Im( E$_{a}$ ) = 1 and for (a) κ 0 = κ$_{a}$ = 1, (b) κ 0 = 1, κ$_{a}$ = √$_{2.}$ .