Exceptional points in quantum and classical dynamics

Abstract

We note that when a quantum system involves exceptional points, i.e. the special values of parameters where the Hamiltonian loses its self-adjointness and acquires the Jordan block structure, the corresponding classical system also exhibits singular behaviour associated with the restructuring of classical trajectories. A system with the crypto-Hermitian Hamiltonian H = (p 2 + z 2 )/2−igz 5 and hyper-elliptic classical dynamics is studied in detail. Analogies with supersymmetric Yang-Mills dynamics are elucidated.

Chaotic systems in complex phase space

“…Our objective here is to generalize (4) into complex phase space and thereby gain a better understanding of the critical behaviour near K c . To accomplish this we are motivated to extend the analysis of refs [3][4][5][6][7][8][9][10][11][12][13][14][15][16] to time-dependent systems. Thus, we treat p n , θ n and sometimes K as complex variables, which we separate into real and imaginary parts as…”

“…In particular, solutions to Hamilton's equations have been examined for various systems whose Hamiltonians are PT symmetric. For such systems the classical trajectories are typically complex [3][4][5][6][7][8][9][10][11][12][13][14][15][16]. These trajectories can lie in many-sheeted Riemann surfaces and often have elaborate topological structure.…”

Conditional observability versus self-duality in a schematic model

“…We will prefer calling our H quasi-Hermitian (a term typical for nuclear physics [7]). Still, several authors also use the equivalent names of pseudo-Hermitian H (predominantly in the context of linear algebra and mathematics [8]) or cryptoHermitian H (this nice and most self-explanatory concept appeared recently in the context of gauge models [9]). Last but not least, another, less strict nickname of PT −symmetric H as coined by Carl Bender [10] became most popular as particularly appealing in field theory.…”

RG-Whitham dynamics and complex Hamiltonian systems

“…However the second step concerning the perturbation and identification of the condensates is more complicated and we shall restrict ourselves by the few conjectures. Note that the previous discussion of the Hamiltonian interpretation of the AD points can be found in [8] however that paper was focused at another aspects of the problem.…”