Operation near non-Hermitian (NH) degeneracies, known as exceptional points (EPs), can lead to enhanced sensitivities with magnitude depending upon the order of the EP. The signature of these anomalies is reflected in the full coalescence of both eigenvalues and eigenvectors. A natural question arises as to how the sensitivity alters if the eigenvectors partially coincide—we shall refer to these anomalies as partial degeneracies (PDs). Here, we derive the maximum sensitivity at these points and the conditions for it to emerge (our analysis also yields the conditions leading to maximum sensitivity at EPs). The investigation begins with the familiar case of EPs and is systematically extended to scenarios where PDs occur within NH settings involving a full or a partial eigenvalue collapse. In this regard, the theory of compound/higher adjugate matrices is employed, while certain results are also verified via the Faddeev-LeVerrier algorithm. Overall, our analysis sheds light on the behavior of NH settings close to degeneracies of any type, resolving a critical limitation of time-independent perturbation theory in such contexts. Moreover, it lays the groundwork for further explorations on the topological features and transport effects associated with evolution near PDs.

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• Geometric & topological phases
• Mathematical physics
• Open quantum systems
• Exceptional points
• Non-Hermitian systems
• Perturbative methods