6 point, which are revealed as the saturation of the wave packet amplitude growth to a steady-state (non- decaying) value. If the lattice is realized in a periodic dielectric medium as discussed in [6], at light wavelength λ = 1 . 5 µ m, assuming a bulk refractive index n$_{s}$ = 1 . 5 and a maximum refractive index change (both real and imaginary parts) of Δ n = 2 × 10 − $^{4}$, the simulations of Fig.2 correspond, as an example, to an optical lattice with spatial period a ≃ 6 . 2 µ m excited by a Gaussian beam of size w ≃ 493 µ m; the spatial units along the x and t axes are l$_{x}$ 6 . 2 µ m and l$_{t}$ ≃ 477 µ m, respectively. The same scales hold for the simulations shown in Figs.4 and 5. FIG. 5: (color online) Same as Fig.4, but above the PT symmetry-breaking point ( λ = 1 . 1). -1200 1200 0 t x (a) (b) t GLYPH<c121> m (t) 0 70 0 50 100 2 4 6 8 10 12 IV. CONCLUSIONS In this work, it has been shown that Bragg scatter- ing in complex crystals provides a physically important process to visualize the appearance of spectral singular- ities in non-Hermitian Hamiltonians with complex peri- odic potentials. In particular, clear signatures of spectral singularities could be gained in a diffraction experiment with wave packets. These results may suggest the in- vestigation of possible experimental systems to observe spectral singularities in complex lattices, such as pho- tonic systems [6, 9]. Fig.4) and above ( λ = 1 . 1 λ$_{c}$ = 1 . 1, Fig.5) the PT symmetry breaking transition point. In both cases, H is diagonalizable and there are not spectral singularities. Note that below the PT symmetry breaking point, the energy spectrum is real-valued and, after an initial increase, ψ$_{m}$ ( t ) does not settle down to a steady-state value, rather it tends to monotonically decay (Fig.4) like in an ordinary (Hermitian) crystal. On the other hand, above the PT symmetry-breaking point ( λ = 1 . 1), the energy spectrum contains pairs of complex-conjugate eigenvalues, and ψ$_{m}$ ( t ) monotonically increases, as shown in Fig.5. Therefore, the scattering process with wave packets can be used to highlight the appearance of spectral singularities at the PT symmetric breaking [1] C.M. Bender, Rep. Prog. Phys. 70 , 947 (2007). [2] A. Mostafazadeh, arXiv:0810.5643. [3] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80 , 5243 (1998). [4] C. M. Bender, D.C. Brody, and H. F. Jones, Phys. Rev. Lett. 89 , 270401 (2002); 92 , 119902(E) (2004). [5] A. Ruschhaupt, F. Delgado, and J.G. Muga, J. Phys. A 38 , L171 (2005). [6] K.G. Makris, R. El-Ganainy, D.N. Christodoulides, and Z.H. Musslimani, Phys. Rev. Lett. 100 , 103904 (2008). [7] S. Klaiman, U. G¨ unther, and N. Moiseyev, Phys. Rev. Lett. 101 , 080402 (2008). [8] O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Phys. Rev. Lett. 103 , 030402 (2009). [9] A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103 , 093902 (2009). [10] M.V. Berry, Czech. J. Phys. 54 , 1039 (2004); W.D. Heiss, J. Phys. A 37 , 2455 (2004). [11] M. A. Naimark, Amer. Math. Soc. Transl. 16 , 103 (1960); V. E. Lyance, Amer. Math. Soc. Transl. 60 , 185 (1967). [12] B. F. Samsonov, J. Phys. A 38 , L397 (2005). [13] A. Mostafazadeh and H. Mehri-Dehnavi, J. Phys. A 42 , 125303 (2009). [14] I. Rotter and A.F. Sadreev, Phys. Rev. E 71 , 036227 (2005); I. Rotter, Czech. J. Phys. 55 , 1167 (2005). [15] E. Persson, I. Rotter, H.-J. St¨ ockmann, and M. Barth, [28] S. K. Godunov, Ordinary differential equations with con- [27] See, for instance, O.A. Veliev, Russian J. Math. Phys. 13 , 101 (2006) and references therein. [26] A.A. Cottey, Am. J. Phys. 39 , 1235 (1971). [25] S. Longhi, Phys. Rev. Lett. 103 , 123601 (2009). [24] L.A. Pastur and V.A. Tkachenko, Math. Notes 50 , 1045 (1991); K.C. Shin, J. Phys. A 37 , 8287 (2004). [23] H.F. Jones, Phys. Lett. A 262 , 242 (1999); Z. Ahmed, Phys. Lett. A 286 , 231 (2001); J.M. Cervero, Phys. Lett. A 317 , 26 (2003); A. Khare and U. Sukhatme, J. Math. Phys. 47 , 062103 (2006); T. Curtright and L. Mezincescu, J. Math. Phys. 48 , 092106 (2007). [22] C.M. Bender, G.V. Dunne, P.N. Meisinger, Phys. Lett. A 252 , 272 (1999). [21] M.V. Berry and D.H.J. O’Dell, J. Phys. A 31 , 2093 (1998). [20] M.V. Berry, J. Phys. A 31 , 3493 (1998). Phys. Rev. Lett. 85 , 2478 (2000); P. Cejnar, S. Heinze, and M. Macek, Phys. Rev. Lett. 99 , 100601 (2007); H. Cartarius, J. Main, and G. Wunner, Phys. Rev. Lett. 99 , 173003 (2007). [16] I. Rotter, J. Phys. A 55 , 1167 (2005). [17] A. Mostafazadeh, Phys. Rev. Lett. 102 , 220402 (2009). [18] A. Mostafazadeh, Phys. Rev. A 80 , 032711 (2009). [19] C. Keller, M.K. Oberthaler, R. Abfalterer, S. Bernet, J. Schmiedmayer, and A. Zeilinger, Phys. Rev. Lett. 79 , 3327 (1997).