0$ such that $$ {\rm Im}\,\lambda_k^\pm(g,p)\neq 0, \quad 1/2-\eta \leq p\leq 1/2 $$ It follows (see e.g.\cite{BS}, \cite{Ea}) that the complex arcs ${\mathcal E}^\pm_k:={\rm Range}(\lambda_k^\pm(g,p)): p\in[1/2-\eta(g), 1/2]$ belong to the spectrum of ${ K}(g)$. This concludes the proof of the theorem. %%%%%%%%%%% %%%%%%%%%%%% %\newpage \vskip 1.5cm\noindent \begin{thebibliography}{DGHLLL} %\vskip 0.5cm\noindent {\small \bibitem[AGHKH]{AGHKH} S.Albeverio, F.Gesztesy, R.H\/eght-Krohn, H.Holden {\it Solvable models in quantum mechanics} , Springer-Verlag 1988 \bibitem[Ah]{Ah} Z.Ahmed, {\it Energy band structure due to a complex, periodic, $PT$ invariant potential}, Phys.Lett. A {\bf 286}, 231-235 (2001) \bibitem[Be1]{Be1} C. M. Bender, S. Boettcher, and P. N. Meisinger, {\it PT-Symmetric Quantum Mechanics} Journal of Mathematical Physics {\bf 40},2201-2229 (1999) \bibitem[Be2]{Be4} C.M.Bender, {\it Making Sense of Non-Hermitian Hamiltoniians} (hep-th/0703096) \bibitem[BBM]{Be2} C. M. 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