Conditional observability versus self-duality in a schematic model

For a 4-dimensional 3-parametric toy Hamiltonian H(a, b, c) we construct the domain D of couplings in which the eigenvalues En remain real (i.e., in principle, observable). A relationship is found between the reflection symmetry of the spectrum (i.e., its Dunne’s and Shifman’s self-duality Ej = −EN+1−j at N = 4) and a geometric symmetry of the physical domain D. Simultaneously, both remain unbroken at a = c.