Third-order spectral branch points in Krein space related setups: $$\mathcal{P}\mathcal{T}$$ -symmetric matrix toy model, MHD α 2-dynamo and extended Squire equation

Abstract

The spectra of self-adjoint operators in Krein spaces are known to possess real sectors as well as sectors of pair-wise complex conjugate eigenvalues. Transitions from one spectral sector to the other are a rather generic feature and they usually occur at exceptional points of square root branching type. For certain parameter configurations two or more such exceptional points may happen to coalesce and to form a higher-order branch point. We study the coalescence of two square root branch points semi-analytically for a PTsymmetric 4×4 matrix toy model and illustrate its occurrence numerically in the spectrum of the 2 × 2 operator matrix of the magneto-hydrodynamic α 2 -dynamo and an extended version of the hydrodynamic Squire equation. Krein space related physical setups and spectral phase transitionsSome basic spectral properties of the Hamiltonians of PT -symmetric Quantum Mechanics (PTSQM) [1] can be easily explained from the fact [2, 3] that these Hamiltonians are self-adjoint operators in Krein spaces [4, 5] -Hilbert spaces with an indefinite metric structure. In contrast to the purely real spectra of selfadjoint operators in "usual" Hilbert spaces (with positive definite metric structure), the spectrum of self-adjoint operators in Krein spaces splits into real sectors and sectors with pair-wise complex conjugate eigenvalues. In physical terms these two types of sectors are equivalent to phases of exact PT -symmetry and spontaneously broken PT -symmetry [1]. The reality of the spectrum of a PTSQM Hamiltonian, e.g., with complex potential ix 3 , means that this spectrum is located solely in a real sector and that the operator is quasi-Hermitian 1 ) in the sense of Ref. [7]. In * ) Presented at the 3rd International Workshop "Pseudo-Hermitian Hamiltonians in Quantum Physics", Istanbul, Turkey, June 20-22, 2005. 1 ) Because quasi-Hermitian operators form only a restricted subclass of self-adjoint operators in Krein spaces (pseudo-Hermitian operators in the sense of Ref.[6]), the question for the reality of the spectrum seems up to now to be only partially solved. Apart from the requirement for existing PT -symmetry of the differential expression of the operator, a subtle interplay between the operator domain and some, in general not yet sufficiently clearly identified, additional structural aspects of its differential expression seems to be responsible for the quasi-Hermiticity of the operator (exact PT -symmetry of the corresponding PTSQM setup).

Why dynamos are prone to reversals

“…Those transition points, which have been found in many dynamo models [36,37], are well known in operator theory as spectral branch points -"exceptional points" of branching type of non-selfadjoint operators [38]. Such branch points are characterized not only by coalescing eigenvalues but also by a coalescence of two or more (geometric) eigenvectors and the formation of a non-diagonal Jordan block structure with associated vectors (algebraic eigenvectors) [39,40,41]. This is in contrast to "diabolical points" [42] which are exceptional points of an accidential crossing of two or more spectral branches with an unchanged diagonal block structure of the operator and without coalescing eigenvectors [38,40].…”

A non-Hermitian \mathcal{P}\mathcal{T} symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

“…The occurrence of EPs in a system has drastic effects on the systems behavior, especially concerning adiabatic features and geometric phases. For the occurrence of EPs in various physical models see, for example and not aiming at any completeness [1,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]. In the theory of PT −symmetric quantum systems EPs naturally occur as phase-transition points between sectors of exact PT −symmetry and sectors of spontaneously broken PT −symmetry.…”

Effects of an exceptional point on the dynamics of a single particle in a time-dependent harmonic trap

“…In addition, theoretical studies predict significant effects of second-order EPs on the photoionization of atoms [19][20][21] and the photodissociation of molecules [22][23][24].The possibility of higher-order EPs (where more than two eigenstates of the non-Hermitian Hamiltonian coalesce at the EP) has been discussed in the literature for time independent PT symmetric Hamiltonians (see Ref. [25,26] and references therein). The main effect of EPs (of any order) on the dynamics of PT -symmetric systems is the sudden transition from a real spectrum to a complex energy spectrum associated with gain and loss processes.…”