New quasi-exactly solvable Hermitian as well as non-Hermitian $$mathcal{P}mathcal{T}$$-invariant potentials

We start with quasi-exactly solvable (QES) Hermitian (and hence real) as well as complex $ mathcal{P}mathcal{T} $ mathcal{P}mathcal{T} -invariant, double sinh-Gordon potential and show that even after adding perturbation terms, the resulting potentials, in both cases, are still QES potentials. Further, by using anti-isospectral transformations, we obtain Hermitian as well as $ mathcal{P}mathcal{T} $ mathcal{P}mathcal{T} -invariant complex QES periodic potentials. We study in detail the various properties of the corresponding Bender-Dunne polynomials.