  We show that at least the quasi-exactly solvable eigenvalues of the
Schr\"odinger equation with the potential $V(x) = -(\zeta \cosh 2x -iM)^2$ as
well as the periodic potential $V(x) = (\zeta \cos 2\theta -iM)^2$ 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.
