Broken spectral reflection symmetry in a solvable PT −symmetric model

Any non-Hermitian PT −symmetric quantum Hamiltonian H remains physical in a domain of parameters D where the spectrum is real and, hence, measurable. It has been conjectured and verified, recently, that the postulate of a self-duality of the spectrum Ej = −EN+1−j could simplify the structure of the manifold D. An independent constructive support of the plausibility of such a relationship is given here via a three-parametric counterexample. A complicated shape of the boundary ∂D is found and described by non-numerical formulae in a non-self-dual “benchmark” matrix model.