4 C. Spectral singularities: the Fano diagonalization procedure Let us assume that the spectrum of the non-Hermitian Hamiltonian H = H$_{0}$ + V is real-valued. As shown in the previous subsection, this implies that the spectrum of H is purely continuous and spans the interval ( E$_{1}$, E$_{2}$ ). Let us indicate by |E〉 the (improper) eigenfunction of H corresponding to the eigenvalue E , and by |E $^{†}$〉 the (im- proper) eigenfunction of the adjoint H † corresponding to the same eigenvalue E . For H to be diagonalizable, the set of functions {|E〉 , |E $^{†}$〉} must form a complete biorthonormal basis of Hilbert space [9], that is on the segment ( E$_{1}$, E$_{2}$ ) of the real axis (as for the self- energy Σ). According to Eq.(15), we may therefore limit to consider the singularities of G $_{a,a}$( z ). From Eqs.(6), (11) and (12) one can conclude that: (i) The continuous spectrum of H is the same as that of H$_{0}$ , i.e. the interval ( E$_{1}$, E$_{2}$ ) of the real axis, where G ( z ) has a branch cut. (ii) The point spectrum of H , corresponding to bound states outside the continuum, are the complex roots z of the equation z − E$_{a}$ = Σ( z ) (18) 〈E $^{†}$| E $^{′}$〉 = δ ( E − E $^{′}$) , ∫ 2 E 1 d E|E〉〈E $^{†}$| = I . (21) A spectral singularity at E = E 0 sets in when 〈E 0 |E 0 〉 = 0 , (22) Im( E$_{a}$ ) = ± πV ( E $_{0}$) (19) E 0 − Re( E$_{a}$ ) = Δ( E $_{0}$) (20) which prevents H to be diagonalizable. To determine the onset of spectral singularities of the FFA Hamiltonian H , we can thus proceed by calculating the eigenfunctions |E〉 of H and |E $^{†}$〉 of its adjoint H † following the diagonal- ization procedure used by Fano [18] in the problem of atomic autoionizing resonances (see also [17, 21]), prop- erly modified to account for the non-Hermitian nature of H . To this aim, we expand the eigenstates |E〉 and |E $^{†}$〉 on the orthonormal and complete basis {| a 〉 , | k 〉} as |E〉 = α ( E ) | a 〉 + ∫ dkβ ( E , k ) | k 〉 (23) |E $^{†}$〉 = α $^{†}$( E ) | a 〉 + dkβ $^{†}$( E , k ) | k 〉 (24) with expansion coefficients α ( E ), α $^{†}$( E ), β ( E , k ) and β $^{†}$( E , k ) to be determined. Note that 〈E $^{′}$| E $^{†}$〉 = α $^{∗}$( E $^{′}$) α $^{†}$( E ) + ∫ dkβ $^{∗}$( E $^{′}$, k ) β $^{†}$( E , k ) . (25) Using Eqs.(1), (2) and (23), the eigenvalue equation H |E〉 = E|E〉 yields the following coupled equations for the expansion coefficients α ( E ) and β ( E , k ): ( E$_{a}$ − E ) α ( E ) + ∫ dkv ( k ) β ( E , k ) = 0 (26) E ( k ) − E ] β ( E , k ) + v $^{∗}$( k ) α ( E ) = 0 . (27) Equation (27) can be solved for β ( E , k ) and substituted into Eq.(26). However, there is a singularity at E ( k ) = E , so that 1 / [ E ( k ) − E ] must be written as its principal and delta function parts, yielding (see [18, 21]) The coefficient z ( E ) in front of the delta function on the right hand side of Eq.(28) is determined by substituting β ( E , k ) = − v $^{∗}$( k ) α ( E ) ( P 1 E ( k ) − E + z ( E ) δ ( E ( k ) − E ) . (28) provided that Im( E$_{a}$ ) = 0 [41]. Note that real-valued energies can not belong to the point spectrum of H because Eq.(18) does not have real-valued roots whenever H is non-Hermitian [42]. This means that the point spectrum of H , if not empty, is strictly complex- valued. On the other hand, the continuous part of the spectrum is real-valued according to the property (i). Therefore, we may conclude that the energy spectrum of the non-Hermitian FFA Hamiltonian is real-valued if and only if its spectrum is purely continuous. In this case, spectral singularities in the continuous spectrum occur whenever Eqs.(19) and (20) can be simultaneously satisfied. It should be noted that the behavior of G ( z ) in the neighborhood of the spectral singularity z = E 0 on the continuous spectrum is different for an ’absorb- ing’ [Im( E$_{a}$ ) < 0] and for an ’amplifying’ [Im( E$_{a}$ ) > 0] complex energy E$_{a}$ of state | a 〉 . Since the spectral cou- pling V ( E $_{0}$) is always positive, for an absorbing complex energy Eq.(19) can be satisfied by taking the lower (neg- ative) sign on the right hand side; correspondingly, from Eqs.(6) and (12) it follows that G $_{a,a}$( z ) is unbounded when z → E 0 from the bottom of the real energy axis, i.e. for Im( z ) < 0, but G $_{a,a}$( z ) remains bounded when z → E 0 with Im( z ) > 0 [43]. Conversely, for an ’amplify- ing’ complex energy Im( E$_{a}$ ) > 0, Eq.(19) can be satisfied by taking the upper (positive) sign on the right hand side; correspondingly, G $_{a,a}$( z ) is unbounded when z → E 0 with Im( z ) > 0, but it remains bounded when z 0 with Im( z ) < 0 [43]. Such a different behavior of spectral singularities for an absorbing or an amplifying complex energy E$_{a}$ has some relevant physical implications, that will be discussed in Sec.II.D and Sec.III.C. In Sec.II.D it will be shown that the appearance of a spectral singular- ity in the amplifying case is responsible for the non-decay of state | a 〉 into the continuum, in spite of the absence of bound states; in Sec.III.C the interplay between spec- tral singularities and wave scattering will be investigated for a semi-infinite tight-binding lattice realization of the FFA Hamiltonian. in correspondence of which the resolvent G ( z ) has a pole. (iii) A spectral singularity E 0 of the continuous spectrum( E$_{1}$, E$_{2}$ ) is any solution of the coupled equa- tions