Asymptotic non-Hermitian degeneracy phenomenon and its exactly solvable simulation

Abstract

Up to these days, the popular PT-symmetric imaginary cubic oscillator did not find any consistent probabilistic quantum-mechanical interpretation because its Hamiltonian has been shown, by mathematicians, intrinsic-exceptional-point (IEP) singular. In the paper we explain why there is even no reasonable small-perturbation-based regularization of the similar unacceptable (i.e., IEP-singular) quantum models. The explanation is based on a partial formal analogy of the IEP singularity with the conventional exceptional point (EP). What is important is that we are able to construct a simplified $N$ by $N$-matrix (and exactly solvable) toy-model Hamiltonian admitting the asymptotic (i.e., high-excitation) EP-related wave-function degeneracy which, in some sense (i.e., in the limit of large $N$) mimics several aspects of its IEP analogue. In this comparison, the difference is that the regularization of the EP singularities is possible (using an ad hoc perturbation of size ${\cal O}(1/N)$) while an analogous regularization of the IEP singularity is not (we have to consider $N \to \infty$).