Fig. 1: Perturbation analysis of defective Hamiltonians (\({\hat{H}}_{{{{{{\rm{EP}}}}}}}\)) and its subtleties at exceptional surfaces.
From: Linear response theory of open systems with exceptional points
a Two crucial steps are involved: (1) finding a perturbation (\({\hat{H}}_{{{{{{\rm{pt}}}}}}}\)) that removes the degeneracy, and (2) obtaining the limit when the perturbation vanishes. \(\hat{G}(\epsilon )\) is the Green operator. The first step can be straightforward in simple systems. b In systems that exhibit exceptional hypersurfaces in the parameter space23, 104, 125,126,127, finding such a perturbation can be a very complex task since any perturbation that shifts the system along the surface will fail. In addition, taking the limit when ϵ → 0 involves the cancellation of several singular terms with opposite signs. For complex geometries with a large number of degrees of freedom, this can be a daunting task. In general, it is not possible a priori to confirm if this approach gives exact answer or approximate response function.
