Perturbation Theory Near Degenerate Exceptional Points

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 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.

Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks

“…Such an input information will be carried by the preselected N by N BH-type toy-model Hamiltonian matrix H [N] (γ), tractable as a small perturbation of its EPN limit H [N] (γ (EPN) ). Subsequently, in a way explained in [19], the γ−independent transition matrices Q can be still perceived as certain formal analogues of the unperturbed basis [23,24].…”

A Mini-Review of the Kinetic Energy Partition Method in Quantum Mechanics

“…Moreover, things become even more challenging when the magnitudes or types of the two potentials are comparable, making it fundamentally impossible to determine which one dominates over the other. At this stage, specialized techniques, such as the large-order perturbation theory [6], strong perturbation theory [8], or re-summation techniques are typically employed. However, while these methods may improve certain results, they often introduce additional problems.…”

Special Issue: “Symmetries in Quantum Mechanics and Statistical Physics”

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