Complex symmetric Hamiltonians and exceptional points of order four and five

Abstract

A systematic elementary linear-algebraic construction of non-Hermitian Hamiltonians H = H(γ) possessing exceptional points γ = γ (EP ) of higher orders is proposed. The implementation of the method leading to the EPs of orders K = 4 and K = 5 is described in detail. Two distinct areas of applicability of our user-friendly benchmark models are conjectured (1) in quantum mechanics of non-Hermitian systems, or (2) in their experimental simulations via classical systems (e.g., coupled waveguides).Keywords non-Hermitian quantum dynamics; multilevel degeneracies; exceptional points of higher orders; non-quantum simulations; coupled waveguides; 1

Confluences of exceptional points and a systematic classification of quantum catastrophes

“…Thirdly, the well known numerically ill-conditioned nature of the study of the limiting transition often forces us to use certain truly sophisticated construction methods in a way sampled, say, in Ref. 70 .…”

Confluences of exceptional points and a systematic classification of quantum catastrophes

“…Thirdly, the well known numerically ill-conditioned nature of the study of the limiting transition g → g (EP ) often forces us to use certain truly sophisticated construction methods in a way sampled, say, in Ref. [68]. For all of these reasons it will make sense to turn attention to the more general matrix models in which the practical calculations remain feasible but in which it should still be possible to enhance the flexibility of the picture of the EP-related dynamics.…”

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

“…In spite of the manifest non-Hermiticity of the P T −symmetric candidates H for Hamiltonians, these operators were shown eligible as generators of unitary evolution [12,14]. In this context, one of the basic methodical assumptions accepted in the current literature on BH models [19,20,[34][35][36][37] was that the infinite-dimensional matrix (5) as well as all of its separate submatrices (6) had to be complex symmetric, tridiagonal and P T −symmetric, with P equal to an antidiagonal unit matrix, and with symbol T representing an antilinear operation of Hermitian conjugation (i.e., transposition plus complex conjugation). This led to the conclusion (or rather conjecture) that in the EPN limit (i.e., at the instant of the loss of diagonalizability), the canonical representation of every N by N submatrix H (N) (γ (EP) ) can be given the form of the N by N Jordan matrix (16) with, due to P T −symmetry, η = 0.…”