2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/57840Hill-determinant method is described and shown applicable within the so called PT-symmetric quantum mechanics. We demonstrate that in a way paralleling its traditional Hermitian applications and proofs the method guarantees the necessary asymptotic decrease of wave functions as resulting from a fine-tuned mutual cancellation of their asymptotically growing exponential components. Technically, the rigorous proof is needed/offered that in a quasi-variational spirit the method allows us to work, in its numerical implementations, with a sequence of truncated forms of the rigorous Hill-determinant power series for the normalizable bound states.review lecture for Proceedings ``Recent Trends in Exponential Asymptotics" (June 28 - July 2, RIMS, Kyoto), 14 pMathematical Physics81Q05text