Exceptional points of the eigenvalues of parameter-dependent Hamiltonian operators

We calculate the exceptional points of eigenvalues several parameter-dependent Hamiltonian operators mathematical and physical interest. show that calculation is greatly facilitated by application discriminant to secular determinant. In this way, problem reduces finding roots a polynomial function just one variable, parameter in operator. As illustrative examples, we consider particle one-dimensional box with potential, periodic Mathieu equation, Stark effect polar rigid rotor symmetric top.