Solvable models in quasi-Hermitian quantum mechanics

We consider an one-dimensional one-parametric PT-symmetric operator Hα, for which we study the existence of the similarity transformation to a self-adjoint operator in dependence on the real parameter α. We will derive for which values of the parameter the model is quasi-Hermitian and for such values we find corresponding similar self-adjoint operator in the closed form. In the cases when the similar self-adjoint operator does not exist we will construct a generalized similarity transformation and find the similar operator in the closed form likewise. Followingly, we will compare these two similar operators and discuss their differences in dependence on the parameter.