Complex symmetric Hamiltonians and exceptional points of order four and five

In the broad context of physics ranging from classical experimental optics to quantum mechanics of unitary as well as non-unitary systems there emerge interesting phenomena related to the presence of the so called Kato's exceptional points in the space of parameters. An elementary linear-algebraic method of their localization is proposed and applied to the class of tridiagonal $N$ by $N$ complex symmetric toy-model generators of evolution $H=H(\gamma)$. The implementation of the method is shown to provide new models with the exceptional points $\gamma=\gamma^{(EP)}$ of higher orders. Two distinct areas of applicability are expected to lie (1) in quantum mechanics of non-Hermitian (open as well as closed) systems, and (2) in the experiments using the coupled classical optical waveguides simulating the EP-related effects in the laboratory.