| Summary: | The recent growth in popularity of the non-Hermitian quantum Hamiltonians <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> with real spectra is strongly motivated by the phenomenologically innovative possibility of an access to the non-Hermitian degeneracies called exceptional points (EPs). What is actually presented in the present paper is a perturbation-theory-based demonstration of a fine-tuned nature of this access. This result is complemented by a toy-model-based analysis of the related details of quantum dynamics in the almost degenerate regime with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≈</mo><msup><mi>λ</mi><mrow><mo>(</mo><mi>E</mi><mi>P</mi><mo>)</mo></mrow></msup></mrow></semantics></math></inline-formula>. In similar studies, naturally, one of the decisive obstacles is the highly nontrivial form of the underlying mathematics. Here, many of these obstacles are circumvented via several drastic simplifications of our toy models—i.a., our <i>N</i> by <i>N</i> matrices <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are assumed real, tridiagonal and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetric, and our <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>H</mi><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is assumed to be split into its Hermitian and non-Hermitian components staying in interaction. This is shown to lead to several remarkable spectral features of the model. Up to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>8</mn></mrow></semantics></math></inline-formula>, their description is even shown tractable non-numerically. In particular, it is shown that under generic perturbation, the “unfolding” removal of the spontaneous breakdown of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">PT</mi></semantics></math></inline-formula>-symmetry proceeds via intervals of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>λ</mi></semantics></math></inline-formula> with complex energy spectra.
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