Abstract
Using the “exact WKB method” [E. Delabaere et al., J. Math. Phys. 38 (1997) 6126], we prove some reality results on the spectrum of some families of non-Hermitian Hamiltonians having PT -symmetry. This partially solves a conjecture of Zinn-Justin and Bessis. In part II [Phys. Lett. A 250 (1998) 29] we will prove non-reality results for other such families.
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