$\mathcal P\mathcal T$ Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues

We prove the reality of the perturbed eigenvalues of some $\mathcal P\mathcal T$ symmetric Hamiltonians of physical interest by means of stability methods. In particular we study 2-dimensional generalized harmonic oscillators with polynomial perturbation and the one-dimensional $x^2(ix)^\epsilon$ for $-1<\epsilon<0$.