2 FFA Hamiltonian has a real-valued energy spectrum if and only if its spectrum is purely continuous. We are interested here to determine, if any, the appearance of spectral singularities of H , which would prevent H to be diagonalizable. This problem can be addressed in two ways: (i) by the determination of the resolvent G ( z ), and (ii) by a direct diagonalization of H following the original procedure by Fano [18], extended to account for the non-Hermitian nature of H . In the former case, a spectral singularity at E = E $_{0}$, embedded in the continu- ous spectrum ( E$_{1}$,E$_{2}$ ), is revealed as a divergence of the Green function G ( x,y ; z ) = 〈 x | G ( z ) y 〉 in a neighborhood of z = E $_{0}$, divergence which does not corresponds to a bound state embedded in the continuum [9]. In the lat- ter case, a spectral singularity at E = E 0 occurs when 〈E † 0 |E 0 〉 = 0, where |E 0 〉 and |E † 0 〉 are the eigenfunctions of H and of its adjoint H $^{†}$, respectively, corresponding to the eigenvalue E 0 [9]. The standard FFA model describes the interaction of a discrete state | a 〉 , of energy E$_{a}$ , with a continuous set of states | k 〉 with energy E ( k ) (see, for instance, [22, 24–26, 29, 36]). Here | a 〉 and | k 〉 represent a com- plete set of Dirac states in the Hilbert space which sat- isfy the orthonormal conditions 〈 a | a 〉 = 1, 〈 a | k 〉 = 0 and 〈 k $^{′}$| k 〉 = δ ( k − k $^{′}$). The Hamiltonian of the full system can be written as H = H$_{0}$ + V , where sions are outlined. A. The model II. SPECTRAL SINGULARITIES IN A NON-HERMITIAN FRIEDRICHS-FANO-ANDERSON MODEL H$_{0}$ = E$_{a}$ | a 〉〈 a | + ∫ dk E ( k ) | k 〉〈 k | (1) V = ∫ dk [ v ( k ) | a 〉〈 k | + v $^{∗}$( k ) | k 〉〈 a | ] (2) B. Spectral singularities: the resolvent approach is the Hamiltonian of the non-interacting discrete and continuous states, and is the (self-adjoint) interaction term, described by the spectral coupling function v ( k ). Typically, we assume that the energy E ( k ) of continuous states spans the in- terval E$_{1}$ < E < E$_{2}$ (eventually E$_{2}$ = ∞ ), and E ( k ) is a monotonic function (either increasing or decreasing) of k , i.e. we assume that there are not energy degenera- cies of continuous states. The state vector of the system | ψ 〉 evolves according to the Schr¨ odinger equation (with ℏ = 1) For a given complex number z , the resolvent operator G ( z ) of the Hamiltonian H is defined as i.e. G ( z )( z − H ) = ( z − H ) G ( z ) = I , where I is the iden- tity operator. The knowledge of the resolvent of H , for any z , is equivalent to the knowledge of the set of eigen- functions and eigenvalues of H . In particular, the singu- larities of G ( z ) in the complex plane define the spectrum of H : an eigenvalue E belonging to the point spectrum of H is a pole of G ( z ), whereas the branch cut of G ( z ) de- termines the continuous part of the spectrum of H . The Hamiltonian H is said to have a spectral singularity at E = E $_{0}$, where E 0 belongs to the continuous spectrum of H , if the function G ( z ) = ( z − H ) − $^{1}$, (4) G $_{χ,ϕ}$( z ) = 〈 χ | G ( z ) ϕ 〉 (5) i ∂ | ψ 〉 ∂t = H | ψ 〉 . (3) is unbounded in the neighborhood of z = E $_{0}$, and E 0 does not belong to the point spectrum of H , i.e. it does not corresponds to a bound state embedded in the con- tinuum [9]. In the previous equation, | χ 〉 and | ϕ 〉 are two assigned functions of the Hilbert space; in particular, for | χ 〉 = | x 〉 and | ϕ 〉 = | y 〉 one obtains the coordinate representation of the resolvent G , i.e. the Green func- tion G ( x,y ; z ) = 〈 x | G ( z ) y 〉 . An interesting property of the FFA Hamiltonian is the possibility to calculate the resolvent in a closed form. The procedure to calculate G ( z ) is well known for the Hermitian case (see, for in- stance, [26]), and can be extended mutatis mutandis to the non-Hermitian FFA model considered in this work. As detailed in the Appendix A, the matrix elements of the resolvent G ( z ) on the complete basis {| a 〉 , | k 〉} read explicitly Note that, provided that the energies E$_{a}$ and E ( k ) are real-valued, the Hamiltonian H is Hermitian. The spec- trum of H can be determined by either analyzing the singularities of the resolvent G ( z ) = ( z − H ) − $^{1}$, or by projecting the eigenvalue equation H | ψ 〉 = E | ψ 〉 on the basis {| a 〉 , | k 〉} . As is well known, the continuous spec- trum of H is E$_{1}$ < E < E$_{2}$ , i.e. the same as that of H$_{0}$ , whereas the point spectrum can be either empty or com- posed by a number of discrete eigenvalues, either outside or embedded into the continuous spectrum (see, for in- stance, [22, 39, 40]). We now relax the Hermiticity condition of the FFA model by allowing the ’energy’ E$_{a}$ of the discrete state | a 〉 to be complex valued. However, we will assume that the spectrum of H remains real-valued in spite of the non- Hermiticity of H . The condition for the spectrum of H to remain real-valued will be discussed below in Sec.II.B and corresponds to the absence of bound states, i.e. to an empty point spectrum. This means that a non-Hermitian