Entire functions, PT-symmetry and Voros's quantization scheme

In this paper, A. Avila’s theorem on convergence of the exact quantization scheme of A. Voros is related to the reality proofs of eigenvalues of certain PT-symmetric boundary value problems. As a result, a special case of a conjecture of C. Bender, S. Boettcher and P. Meisinger on reality of eigenvalues is proved.