8 In the Wannier representation, one can readily show that and so that the Hamiltonian H = H$_{0}$ + V can be written in the equivalent form ∫ dkE ( k ) | k 〉〈 k | = − κ$_{0}$ ∑ n =1 ( | n 〉〈 n + 1 | + | n + 1 〉〈 n | ) (61) V = ∫ dk [ v ( k ) | a 〉〈 k | + v $^{∗}$( k ) | k 〉〈 a | ] = − κ$_{a}$ ( | a 〉〈 1 | + | 1 〉〈 a | ) (62) H = − κ$_{0}$ ∞ ∑ n =1 ( | n 〉〈 n + 1 | + | n + 1 〉〈 n | ) + E$_{a}$ | a 〉〈 a | + − κ$_{a}$ ( | a 〉〈 1 | + | 1 〉〈 a | ) . (63) FIG. 3: (color online) Schematic of a semi-infinite one- dimensional tight-binding lattice attached to a boundary im- purity site | a 〉 . Im $^{2}$( E$_{a}$ ) = κ 4 a κ 2 0 [ 1 − Re $^{2}$( E$_{a}$ ) (2 κ$_{0}$ − κ 2 $_{a}$/κ$_{0}$ ) 2 ] . (56) Re( E$_{a}$ ) / (1 − κ $_{a}$/ 2 κ $_{0}$) [see Eq.(55)], is found provided that In its present form, Eq.(63) describes single-particle elec- tron or photon transport on a one-dimensional semi- infinite tight-binding lattice [29, 36, 37], with a hopping rate κ$_{0}$ between adjacent sites of the lattice and with the boundary attached to an impurity site | a 〉 with ’complex’ potential energy E$_{a}$ and with hopping rate κ$_{a}$ (see Fig.3). From a physical viewpoint, the complex potential at the boundary impurity site may account for e.g. loss of the quantum particle flux into other decay channels (quan- tum absorbing potentials [47]), or optical gain or loss of light waves in photonic structures [34, 35]. For instance, light transport in a semi-infinite waveguide array as in Ref.[37], but with a lossy (or active) boundary waveg- uide, provides a simple and experimentally accessible re- alization of the tight-binding model (63). It should be noted that transport and scattering phenomena in tight- binding lattices with complex potentials have been the- oretically investigated in recent works (see, for instance, [47, 48]), however spectral singularities were not found in these previous models. C. Spectral singularities and lattice wave reflection B. Lattice realization provided that Re( E$_{a}$ ) = 0. The physical meaning of such spectral singularities will be discussed in Sec.III.C. It can be readily shown that the condition (56) defines the boundary of the domains shown in Fig.2, i.e. a spectral singularity appears when the boundary of existence of bound states is approached. The case κ$_{a}$/κ$_{0}$ = √$_{2 is}$ somehow singular. From Eqs.(54) and (55), for κ$_{a}$/κ$_{0}$ = √$_{2 it follows that there are}$ two spectral singularities at energies E 0 = ± √ 4 κ 2 0 − Im $^{2}$( E$_{a}$ ) (57) FFA models are often encountered in connection to single-particle electronic or photonic transport in one- dimensional tight-binding lattices or networks (see, e.g., [25–29, 33–38] and references therein), and in most cases the underlying Hamiltonian is Hermitian. In particu- lar, the Hermitian limit of the FFA Hamiltonian H con- sidered in the previous subsection has been previously studied in [29, 36] and shown to be equivalent to a tight- binding Hamiltonian of a semi-infinity lattice with an impurity site. The equivalence can be proven after rep- resenting the Bloch states | k 〉 of the tight-binding energy band in terms of localized Wannier states | n 〉 on a lattice. Let us introduce the Wannier states | n 〉 as The physical meaning of spectral singularities in the non-Hermitian FFA model can be captured by analyzing the wave reflection properties of the lattice of Fig.3. As it will be shown below, a spectral singularity corresponds to the appearance of a diverging peak in the reflectance spectrum when the boundary site is an ’amplifying’ im- purity, i.e. when Im( E$_{a}$ ) > 0, and to the vanishing of wave reflection when the boundary site is an ’absorbing’ impurity, i.e. when Im( E$_{a}$ ) < 0. In the former case, we retrieve for a ’discrete’ scattering problem the physical explanation of spectral singularities as resonances with vanishing spectral width, established by Mostafazadeh Ref.[9] in the framework of ’continuous’ wave scatter- ing by complex potentials. Conversely, the latter case, i.e. that of an ’absorbing’ impurity site, gives a different manifestation of a spectral singularity: a wave incident on the lattice boundary is totally absorbed. for n = 1 , 2 , 3 , ... . Taking into account that | n 〉 = √ 2 π ∫ π 0 dk sin( nk ) | k 〉 (58) ∫ π 0 dk sin( nk ) sin( mk ) = π 2 δ$_{n,m}$ (59) ( n, m ≥ 1), one can readily show that the Wannier states form an orthonormal system, i.e. 〈 n | m 〉 = δ$_{n,m}$ . Addi- tionally, from Eq.(58) it follows that the Bloch states | k 〉 can be decomposed as a superposition of Wannier states | n 〉 according to | k 〉 = √ 2 π ∞ ∑ n =1 sin( nk ) | n 〉 . (60) . . . .. . | a GLYPH<c107>$_{0}$ GLYPH<c107>$_{0}$ GLYPH<c107>$_{0}$ GLYPH<c107>$_{0}$ GLYPH<c107>$_{0}$ GLYPH<c107>$_{a}$ GLYPH<c107>$_{0}$ > |1 > |2 > |3 > GLYPH<c107>$_{0}$ | n > . . . .. .