Eigenvalue sensitivity from eigenstate geometry near and beyond arbitrary-order exceptional points

Systems with an effective non-Hermitian Hamiltonian display an enhanced sensitivity to parametric and dynamic perturbations. I derive a general and exact algebraic expression for this sensitivity that retains a simple asymptotic behaviour close to exceptional points (EPs) of any order, while capturing the role of additional states in the system. This reveals that such states can have a direct effect even if they are spectrally well separated. The employed algebraic approach, which follows the eigenvectors-from-eigenvalues school of thought, also provides direct insights into the geometry of the states near an EP. In particular, I show that the condition number quantifying the sensitivity follows a striking equipartition principle in the quasi-degenerate subspace.