7 Let us consider the non-Hermitian FFA model defined by the following relations for the energy dispersion E ( k ) and spectral coupling v ( k ) where κ$_{0}$ , κ$_{a}$ are two real-valued positive constants and 0 ≤ k ≤ π . The Hermitian limit of this model, at- tained by assuming Im( E$_{a}$ ) = 0, is a special case of the FFA model previously investigated in Ref.[36], which is exactly solvable (see also [29]). Note that the con- tinuous spectrum of H spans the band ( E$_{1}$, E$_{2}$ ), with E$_{2}$ = − E$_{1}$ = 2 κ$_{0}$ . The density of states for this model is given by E ( k ) = − 2 κ$_{0}$ cos k , v ( k ) = − √ 2 π κ$_{a}$ sin k (47) A. The Hamiltonian FIG. 2: Domains of non-existence of bound states for the Hamiltonian H in the (Im( E$_{a}$ ) /κ 0 , κ$_{a}$/κ 0 ) plane (shaded re- gions) for increasing values of the ratio | Re( E$_{a}$ ) | /κ 0 . For a non-Hermitian Hamiltonian, i.e. Im( E$_{a}$ ) = 0, in the shaded regions the energy spectrum of H is real-valued and purely continuous. Spectral singularities occur at the boundary of the shaded regions. κ$_{a}$/κ$_{0}$ ≤ √$_{2 and shrinks toward Im(}$$_{E}$$_{a}$$_{)}$$_{/κ}$$_{0}$ = κ$_{a}$/κ$_{0}$ = 0 as | Re( E$_{a}$ ) /κ$_{0}$ | → 2 $^{−}$. For | Re( E$_{a}$ ) /κ$_{0}$ | ≤ 2, bound states do exist for any value of κ$_{a}$/κ$_{0}$ and Im( E$_{a}$ ) /κ$_{0}$ . The wider domain is attained for Re( E$_{a}$ ) = 0. In particular, for Re( E$_{a}$ ) = 0 and κ$_{a}$/κ$_{0}$ = √$_{2, from Eq.(52) it follows}$ that H has a real-valued energy spectrum provided that − 2 κ$_{0}$ < Im( E$_{a}$ ) < 2 κ$_{0}$. (53) Let us now consider the occurrence of spectral singulari- ties. According to Eqs.(19) and (20) and using Eqs.(49) and (51), a spectral singularity at energy E = E $_{0}$, in- side the interval ( − 2 κ$_{0}$, 2 κ$_{0}$ ), is found provided that the following two equations are simultaneously satisfied 1.6 1 0.6 0.4 0.2 0 0 -1 2 -2 1 1.4 1.2 0.8 0 -1 2 -2 1 1.6 1 0.6 0.4 0.2 0 1.4 1.2 0.8 0 -1 2 -2 1 0 -1 2 -2 1 Im( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$ Im( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$ GLYPH<c107> a / GLYPH<c107> GLYPH<c48> GLYPH<c107> a / GLYPH<c107> GLYPH<c48> Re( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$GLYPH<c61>GLYPH<c48> Re( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$GLYPH<c61>GLYPH<c32>GLYPH<c32>GLYPH<c32>GLYPH<c48>GLYPH<c46>GLYPH<c53> 1.6 1 0.6 0.4 0.2 0 1.4 1.2 0.8 1.6 1 0.6 0.4 0.2 0 1.4 1.2 0.8 Re( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$GLYPH<c61>GLYPH<c32>GLYPH<c32>GLYPH<c32>GLYPH<c49> Re( )/ E$_{a}$ GLYPH<c107>$_{GLYPH<c48>}$GLYPH<c61>GLYPH<c32>GLYPH<c32>GLYPH<c32>GLYPH<c49>GLYPH<c46>GLYPH<c56> + - + - + - ρ ( E ) = ( ∂E ∂k ) − 1 = { 1 √ 4 κ 2 $_{0}$− E 2 − 2 κ$_{0}$ < E < 2 κ$_{0}$ 0 | E | > 2 κ$_{0}$ (48) which shows van-Hove singularities at the band edges, whereas the positive spectral function V ( E ), defined by V ( E ) = ρ ( E ) | v ( E ) | $^{2}$, reads  which is non-singular. Substitution of Eq.(49) into Eq.(11) yields the following expression for the self-energy Σ( z ) [46] V ( E ) =   κ 2 a πκ 0 √ 1 − ( E 2 κ 0 ) 2 − 2 κ$_{0}$ < E < 2 κ$_{0}$ 0 | E | > 2 κ$_{0}$ (49) Σ( z ) = − i κ 2 a 2 κ 2 0 ( √ 4 κ 2 0 − z 2 + iz ) (50) Δ( E ) = Re ( Σ( z = E ± i 0 $^{+}$) ) = (51) =     κ 2 a 2 κ 2 0 ( E + √ E 2 − 4 κ 2 0 ) E < − 2 κ$_{0}$ κ 2 a 2 κ 2 0 E − 2 κ$_{0}$ ≤ E ≤ 2 κ$_{0}$ κ 2 a 2 κ 2 0 ( E − √ E 2 − 4 κ 2 0 ) E > 2 κ$_{0}$ Im( E$_{a}$ ) = ± κ 2 a κ$_{0}$ √ 1 − ( E 0 2 κ$_{0}$ ) 2 (54) Re( E$_{a}$ ) = ( 1 − κ a 2 κ 2 0 ) E $_{0}$. (55) and thus [see Eq.(12)]  2 κ 2 0 0 The condition for the non-Hermitian Hamiltonian to pos- sess a real-valued spectrum (i.e. to avoid complex-valued energies arising from bound states outside the contin- uum) is derived in Appendix B. Precisely, let ξ$_{1}$$_{,}$$_{2}$ be the two roots of the second-order algebraic equation ξ 2 + E$_{a}$ κ$_{0}$ ξ + 1 − ( κ$_{a}$/κ$_{0}$ ) 2 = 0 . (52) For arbitrarily given values of E$_{a}$ , κ$_{a}$ and κ$_{0}$ , the above conditions are generally not satisfied [nowhere for E 0 in the range ( − 2 κ$_{0}$, 2 κ$_{0}$ )], i.e. the non-Hermitian FFA Hamiltonian is generally diagonalizable. Spec- tral singularities appear solely when a constraint among Re( E$_{a}$ ) /κ$_{0}$ , Im( E$_{a}$ ) /κ$_{0}$ and κ$_{a}$/κ$_{0}$ is satisfied. Let us first assume κ$_{a}$/κ$_{0}$ strictly smaller that √$_{2.}$ In this case, a single spectral singularity, at the energy E 0 = Then the Hamiltonian H has a real-valued energy spec- trum if and only if | ξ$_{1}$$_{,}$$_{2}$ | ≤ 1. Figure 2 shows the domain in the plane (Im( E$_{a}$ ) /κ$_{0}$, κ$_{a}$/κ$_{0}$ ) where H has a purely continuous energy spectrum for a few increasing values of the ratio | Re( E$_{a}$ ) /κ$_{0}$ | . The domain lies in the sector