13 roots ξ$_{1}$$_{,}$$_{2}$ of Eq.(B8) satisfy the condition | ξ$_{1}$$_{,}$$_{2}$ | ≤ 1, there are not surface states at the lattice edge. which is Eq.(52) given in the text. Localization of the surface state at the lattice edge requires c$_{n}$ → 0 as n → ∞ , i.e. Re( µ ) > 0 [see Eq.(B4)]. Therefore, if the two [24] P. Lambropoulos, G.M. Nikolopoulos, T.R. Nielsen, and S. Bay, Rep. Prog. Phys. 63 , 455 (2000). [25] G.D. Mahan, Many-Particle Physics (New York, Plenum Press, 1990), pp.272-285. [26] M. Cini, Topics and Methods in Condended-Matter The- ory (Springer, Heidelberg, 2007), Chap.5, pp.81-89. [27] J.W. Gadzuk and M. Plihal, Farady Discuss. 117 , 1 (2000). [28] N. Stefanou and A. Modinos, Phys. Rev. B 57 , 12127 (1998); P. A. Orellana, M. L. Ladr´ on de Guevara, and F. Claro, Phys. Rev. B 70 , 233315 (2004); L. Zhou, F.M. Hu, J. Lu, and C.P. Sun, Phys. Rev. A 74 , 032102 (2006); E. Rufeil Fiori and H.M. Pastawski, Chem. Phys. Lett. 420 , 35 (2006); G.-B. Zhang, S.-J. Wang, and L. Li, Phys. Rev. B 74 , 085106 (2006); A. V. Malyshev, P. A. Orellana, and F. Dom´ınguez-Adame, Phys. Rev. B 74 , 033308 (2006); P. Zedler, G. Schaller, G. Kiesslich, C. Emary, and T. Brandes, Phys. Rev. B 80 , 045309 (2009). [29] S. Tanaka, S. Garmon, and T. Petrosky, Phys. Rev. B 73 , 115340 (2006); S. Tanaka, S. Garmon, G. Ordonez, and T. Petrosky, Phys. Rev. B 76 , 153308 (2007); H. Naka- mura, N. Hatano, S. Garmon, and T. Petrosky, Phys. Rev. Lett 99 , 210404 (2007). [30] T. Petrosky, I. Prigogine and S. Tasaki, Physica A 173 , 175 (1991); G. Ordonez, T. Petrosky, and I. Prigogine, Phys. Rev. A 63 , 052106 (2001). [31] H. Nakazato, M. Namiki, and S. Pascazio, Int. J. Mod. Phys. B 10 , 247 (1996); P. Facchi, H. Nakazato, and S. Pascazio, Phys. Rev. Lett. 86 , 2699 (2001); P. Facchi and S. Pascazio, La Regola dOro di Fermi , in Quaderni di Fisica Teorica, edited by S. Boffi (Bibliopolis, Napoli, 1999). [32] A.G. Kofman and G. Kurizki, Nature (London) 405 , 546 (2000); A.G. Kofman and G. Kurizki, Phys. Rev. Lett. 87 , 270405 (2001); X.-H. Wang, B.-Y. Gu, R. Wang, and H.-Q. Xu, Phys. Rev. Lett. 91 , 113904 (2003); J. Mar- torell, D.W.L. Sprung, W. van Dijk, and J.G. Muga, Phys. Rev. A 79 , 062104 (2009). [33] S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, Phys. Rev. Lett. 80 , 960 (1998); S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manola- tou, and H. A. Haus, Phys. Rev. B 59 , 15882 (1999); P. Chak, S. Pereira, and J. E. Sipe, Phys. Rev. B 73 , 035105 (2006); L.-L. Lin, Z.-Y. Li, and B. Lin, Phys. Rev. B 72 , 165330 (2005). [34] Y. Xu, Y. Li, R. K. Lee, and A. Yariv, Phys. Rev. E 62 , 7389 (2000). [35] S. Longhi, Phys. Rev. A 74 , 063826 (2006). [36] S. Longhi, Eur. Phys. J. B 57 , 45 (2007). [37] S. Longhi, Phys. Rev. Lett. 97 , 110402 (2006); P. Biag- oni, G. Della Valle, M. Ornigotti, M. Finazzi, L. Duo, P. Laporta and S. Longhi, Optics Express 16 , 3762 (2008). [38] A.E. Miroshnichenko, S. Flach, and Y.S. Kivshar, ”Fano resonances in nanoscale structures”, arXiv:0902.3014v4 (2009). [39] G. Sudarshan, in Field Theory, Quantization and Statis- [1] C.M. Bender, Rep. Prog. Phys. 70 , 947 (2007). [2] P. Dorey, C. Dunning, and R. Tateo, J. Phys.A 40 , R205 (2007). [3] A. Mostafazadeh, ”Pseudo-Hermitian Quantum Mechan- ics”, arXiv:0810.5643v2 (2008). [4] C.M. Bender and S. Boettcher, Phys. Rev. Lett. 80 , 5243 (1998). [5] T. Kato, Perturbation Theroy of Linear Operators (Springer, Berlin, 1966). [6] M.V. Berry, Czech. J. Phys. 54 , 1039 (2004). [7] R. R.D. Kemp, Can. J. Math. 10 , 447 (1958); M. A. Naimark, Amer. Math. Soc. Transl. 16 , 103 (1960); J. Schwartz, Commun. Pure Appl. Math. 13 , 609 (1960); V. E. Ljance, Amer. Math. Soc. Trans. 60 , 185 (1967). [8] B. F. Samsonov, J. Phys. A 38 , L397 (2005). [9] A. Mostafazadeh and H. Mehri-Dehnavi, J. Phys. A 42 , 125303 (2009). [10] W. D. Heiss, Phys. Rep. 242 , 443 (1994); E. Narevi- cius and N. Moiseyev, Phys. Rev. Lett. 81 , 2221 (1998); C. Dembowski, H.-D. Gr¨ af, H. L. Harney, A. Heine, W. D. Heiss, H. Rehfeld, and A. Richter, Phys. Rev. Lett. 86 , 787 (2001); C. Dembowski, B. Dietz, H.-D. Graf, H. L. Harney, A. Heine, W. D. Heiss, and A. Richter, Phys. Rev. E 69 , 056216 (2004); T. Stehmann, W. D. Heiss, and F. G. Scholtz, J. Phys. A 37 , 7813 (2004); J. Rubinstein, P. Sternberg, and Q. Ma, Phys. Rev. Lett. 99 , 167003 (2007); U. G¨ unther, I. Rotter, and B. Sam- sonov, J. Phys. A 40 , 8815 (2007); P. Cejnar, S. Heinze, and M. Macek, Phys. Rev. Lett. 99 , 100601 (2007); S. Klaiman, U. G¨ unther, and N. Moiseyev, Phys. Rev. Lett. 101 , 080402 (2008); M. M¨ uller and I. Rotter, J. Phys. A 41 , 244 018 (2008). [11] A. Mostafazadeh, Phys. Rev. Lett. 102 , 220402 (2009). [12] A. Mostafazadeh, J. Phys. A 39 , 13495 (2006). [13] Z. Ahmedar, ”Zero width resonance (spectral singularity) in a complex PT-symmetric potential”, Xiv:0908.2876v1 (2009). [14] J.G. Muga, J.P. Palao, B. Navarro, and I.L. Egusquiza, Phys. Rep. 395 , 357 (2004). [15] A. Mostafazadeh, ”Resonance Phenomenon Related to Spectral Singularities, Complex Barrier Potential, and Resonating Waveguides”, arXiv:0908.1713v1 (2009). [16] A. Ruschhaupt, F. Delgado, and J.G. Muga, J. Phys. A 38 , L171 (2005). [17] K.O. Friedrichs, Commun. Pure Appl. Math. 1 , 361 (1948). [18] U. Fano, Phys. Rev. 124 , 1866 (1961). [19] P.W. Anderson, Phys. Rev. 164 , 41 (1961). [21] P.L. Knight, M.A. Lauder, and B.J. Dalton, Phys. Rep. 190 , 1 (1990). [20] B. Piraux, R. Bhatt, and P.L. Knight, Phys. Rev. A 41 , 6296 (1990). [22] C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions (Wiley, New York, 1992). [23] A.G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41 , 353 (1994).