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Keywords = Hilbert-space geometry near EPs

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17 pages, 336 KiB  
Article
Perturbation Theory Near Degenerate Exceptional Points
by Miloslav Znojil
Symmetry 2020, 12(8), 1309; https://doi.org/10.3390/sym12081309 - 5 Aug 2020
Cited by 9 | Viewed by 3369
Abstract
In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed and described. The motivation of such an extension of the list of the currently available perturbation-approximation recipes was four-fold: (1) its need results [...] Read more.
In an overall framework of quantum mechanics of unitary systems a rather sophisticated new version of perturbation theory is developed and described. The motivation of such an extension of the list of the currently available perturbation-approximation recipes was four-fold: (1) its need results from the quick growth of interest in quantum systems exhibiting parity-time symmetry (PT-symmetry) and its generalizations; (2) in the context of physics, the necessity of a thorough update of perturbation theory became clear immediately after the identification of a class of quantum phase transitions with the non-Hermitian spectral degeneracies at the Kato’s exceptional points (EP); (3) in the dedicated literature, the EPs are only being studied in the special scenarios characterized by the spectral geometric multiplicity L equal to one; (4) apparently, one of the decisive reasons may be seen in the complicated nature of mathematics behind the L≥2 constructions. In our present paper we show how to overcome the latter, purely technical obstacle. The temporarily forgotten class of the L>1 models is shown accessible to a feasible perturbation-approximation analysis. In particular, an emergence of a counterintuitive connection between the value of L, the structure of the matrix elements of perturbations, and the possible loss of the stability and unitarity of the processes of the unfolding of the singularities is given a detailed explanation. Full article
(This article belongs to the Special Issue Symmetries in Quantum Mechanics and Statistical Physics)
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