2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/88825We show that at least the quasi-exactly solvable eigenvalues of the Schrödinger equation with the potential $V(x) = -(ζ\cosh 2x -iM)^2$ as well as the periodic potential $V(x) = (ζ\cos 2θ-iM)^2$ are real for the PT-invariant non-Hermitian potentials in case the parameter $M$ is any odd integer. We further show that the norm as well as the weight functions for the corresponding weak orthogonal polynomials are also real.13 pages, Latex, no figs Revised version, Major changes in Title, Abstract, Introduction and Conclusion; Refs addedQuantum PhysicsHigh Energy Physics - TheoryMathematical Physicstext