symmetry breaking and exceptional points for a class of inhomogeneous complex potentials

We study a three-parameter family of -symmetric Hamiltonians, related via the ODE/IM correspondence to the Perk-Schultz models. We show that real eigenvalues merge and become complex at quadratic and cubic exceptional points, and explore the corresponding Jordan block structures by exploiting the quasi-exact solvability of a subset of the models. The mapping of the phase diagram is completed using a combination of numerical, analytical and perturbative approaches. Among other things this reveals some novel properties of the Bender-Dunne polynomials, and gives new insight into a phase transition to infinitely many complex eigenvalues that was first observed by Bender and Boettcher. A new exactly solvable limit, the inhomogeneous complex square well, is also identified.