Matrix Hamiltonians with an algebraic guarantee of unbroken PT −symmetry

Although the quantum bound-state energies may be generated by the so called PT −symmetric Hamiltonians H = PH†P 6 = H where P is, typically, parity, the spectrum only remains real and observable (i.e., in the language of physics, the PT −symmetry remains unbroken) inside a domain D of couplings. We show that the boundary ∂D (i.e., certain stability and observability horizon formed by the Kato’s exceptional points) remains algebraic (i.e., we determine it by closed formulae) for a certain toy-model family of N−dimensional anharmonic-oscillator-related matrix Hamiltonians H with N = 2, 3, . . . , 11.