| Summary: | In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed and described. The motivation of such an extension of the list of the currently available perturbation-approximation recipes was four-fold: (1) its need results from the quick growth of interest in quantum systems exhibiting parity-time symmetry (<inline-formula><math display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetry) and its generalizations; (2) in the context of physics, the necessity of a thorough update of perturbation theory became clear immediately after the identification of a class of quantum phase transitions with the non-Hermitian spectral degeneracies at the Kato’s exceptional points (EP); (3) in the dedicated literature, the EPs are only being studied in the special scenarios characterized by the spectral geometric multiplicity <i>L</i> equal to one; (4) apparently, one of the decisive reasons may be seen in the complicated nature of mathematics behind the <inline-formula><math display="inline"><semantics><mrow><mi>L</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula> constructions. In our present paper we show how to overcome the latter, purely technical obstacle. The temporarily forgotten class of the <inline-formula><math display="inline"><semantics><mrow><mi>L</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> models is shown accessible to a feasible perturbation-approximation analysis. In particular, an emergence of a counterintuitive connection between the value of <i>L</i>, the structure of the matrix elements of perturbations, and the possible loss of the stability and unitarity of the processes of the unfolding of the singularities is given a detailed explanation.
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