5 according to Fano [18]. The physical meaning of Eq.(39) is well known [18]: owing to the coupling with the con- tinuum, the discrete state | a 〉 is ’diluited’ throughout a set of continuous states (i.e., it becomes a resonance for H ) with a typical resonance curve | α ( E ) | 2 peaked at E ≃ E$_{a}$ − Δ( E$_{a}$ ) of width ≃ πV ( E$_{a}$ ) [44]. For the non-Hermitian FFA model, the energy E$_{a}$ is complex-valued and from Eq.(37) it follows that a spec- tral singularity can appear at the energy E = E 0 of the continuous spectrum such that π 2 + z ∗ $^{2}$( $_{0}$) = 0, i.e. when z $^{∗}$( E $_{0}$) = ∓ iπ. (40) Using Eq.(29), Eq.(40) yields the following conditions for the appearance of a spectral singularity in the non- Hermitian FFA model which are precisely Eqs.(19) and (20) derived in Sec.II.B following the resolvent approach to spectral singularities. Im( E$_{a}$ ) = ± πV ( E $_{0}$) , Re( E$_{a}$ ) − E 0 + Δ( E $_{0}$) = 0 (41) D. Spectral singularities and decay dynamics Hermitian FFA models are often used to describe the evolution of unstable quantum systems and the related problem of quantum mechanical decay and deviations from an exponential decay law (see, for instance, [31, 32] and references therein). If the system is initially prepared in state | a 〉 , i.e. if | ψ ( t = 0) 〉 = | a 〉 , the survival proba- bility P ( t ) to find the system at state | a 〉 at a successive time t is given by P ( t ) = | c$_{a}$ ( t ) | $^{2}$, where c$_{a}$ ( t ) = 〈 a | ψ ( t ) 〉 = 〈 a | exp( − iHt ) | a 〉 . (42) Here we consider the case of a non-Hermitian FFA Hamil- tonian H with a real-valued (and thus purely continuous) energy spectrum, and briefly discuss the physical implica- tions of spectral singularities of H on the decay dynamics of survival probability P ( t ). The temporal evolution op- erator exp( − iHt ) of the system can be written in terms of the resolvent G ( z ) as exp( − iHt ) = i 2 π ∫ B dzG ( z ) exp( − izt ) (43) where the Bromwich path B is any horizontal line Im( z ) = const > 0 in the upper half of the complex z plane [see Fig.1(a)]. Substitution of Eq.(43) into Eq.(42) yields c$_{a}$ ( t ) = i 2 π ∫ B dz G $_{a,a}$( z ) exp( − izt ) (44) where the matrix element G $_{a,a}$( z ) of the resolvent is given by Eq.(6). As discussed in Sec.II.B, G $_{a,a}$( z ) is analytic in the full complex plane, expect for the branch cut on the real axis, from z = E$_{1}$ to z = E$_{2}$ (see Fig.1), where it | α ( E ) | 2 = V ( E ) π $^{2}$V $^{2}$( E ) + [ E$_{a}$ − E + Δ( E )] 2 (39) α $^{∗}$( E ) α $^{†}$( E ) V ( E ) [ π 2 + z ∗ $^{2}$( E ) ] = 1 . (38) For the Hermitian FFA model, the energy E$_{a}$ is real- valued, z ( E ) turns out to be real-valued and α $^{†}$( E ) = α ( E ) [because of |E $^{†}$〉 = |E〉 ], so that Eq.(38) is solved to yield The calculation of the integral on the right hand side of Eq.(34) is complicated by the product of two principal parts, which must be properly resolved into its partial- fraction and singular terms (see [18] or the Appendix of Ref.[21]). Taking into account that F ( E , E $^{′}$) = ∫ E 2 E 1 dEV ( E ) ( P 1 E − E + z $^{∗}$( E ) δ ( E − E ) × ( P 1 E − E ′ + z $^{∗}$( E $^{′}$) δ ( E − E $^{′}$) ) + 1 . (34) β $^{†}$( E , k ) = − v $^{∗}$( k ) α $^{†}$( E ) ( P 1 E ( k ) − E + z $^{∗}$( E ) δ ( E ( k ) − E ) . (32) 〈E $^{′}$| E $^{†}$〉 = α $^{∗}$( E $^{′}$) α $^{†}$( E ) F ( E , E $^{′}$) (33) Substitution of Eqs.(28) and (32) into Eq.(25) yields where we have set and the following expression of the coefficient β $^{†}$( E , k ) can be derived following the same procedure as above where Δ( E ) is defined by Eq.(13), V ( E ) = ρ ( E ) | v ( E ) | 2 and ρ ( E ) = ( ∂E ( k ) /∂k ) − 1 is the density of states. Similarly, the eigenvalue equation H $^{†}$| E $^{†}$〉 = E|E $^{†}$〉 yields ( E ∗ a − E ) α $^{†}$( E ) + ∫ dkv ( k ) β $^{†}$( E , k ) = 0 (30) [ E ( k ) − E ] β $^{†}$( E , k ) + v $^{∗}$( k ) α $^{†}$( E ) = 0 (31) z ( E ) = E$_{a}$ − E + Δ( E ) V ( E ) (29) Eq.(28) into Eq.(26) and requiring that α ( E ) does not vanish. This yields and using Eq.(29), from Eq.(34) one obtains P 1 E − E P 1 E − E ′ = P 1 E − E ′ ( P 1 E ′ − E − P 1 E − E ) + + π $^{2}$δ ( E ′ − E ) δ ( E − E ) (35) F ( E , E $^{′}$) = V ( E ) [ π 2 + z ∗ $^{2}$( E ) ] δ ( E − E $^{′}$) (36) 〈E $^{′}$| E $^{†}$〉 = α $^{∗}$( E ) α $^{†}$( E ) V ( E ) [ π + z $^{2}$( E ) ] δ ( E − E $^{′}$) . (37) so that [see Eq.(33)] In order {|E〉 , |E $^{†}$〉} to represent a complete biorthonor- mal set of functions [Eq.(21)], the amplitudes α ( E ) and α $^{†}$( E ) should be thus normalized such that