Spectral singularities, brachistochrone dilation and the relevance of the Hessenberg type

In the first 1/3 of the talk a brief overview over some mathematical aspects connected with the occurrence of spectral singularities (exceptional points, EPs) will be presented. Based on simple matrix models we discuss stratified manifolds in parameter spaces on which the matrix eigenvalues degenerate. We comment on discriminant sets and similarity relations to canonical Jordan structures, demonstrate the mechanism underlying the formation of self-orthogonal (isotropic) eigenvectors, relate it to corresponding projectors.
In the second 1/3, we sketch the basics of some recent findings on the brachistochrone problem of PT-symmetric quantum mechanics (PTSQM) and its embedding into a setup of standard quantum mechanics (SQM) in a higher dimensional Hilbert space. The embedding uses a Naimark dilation/extension technique for positive operator valued measures (POVMs) built over the nonorthogonal eigenvectors of the PT-symmetric Hamiltonian and its adjoint. We demonstrate that in the SQM model the zero-passage time is preserved for a subsystem which is entangled with a second subsystem of strongly dominant type. Applications towards ultrafast quantum computation processes are hypothesized.
In the last 1/3 of the talk, we sketch the specific mathematical structures underlying the unfolding mechanisms of higher-order exceptional points (EPs) in PT-symmetric Bose-Hubbard models as they can be used for the description of Bose-Einstein condensates (BECs) localized in symmetrically coupled gain/loss potentials. We demonstrate that the basic structure is connected with a nontrivial Jordan block in the spectral decomposition of the Hamiltonian which is perturbed by a small matrix term of a specific upper Hessenberg type. The concrete Hessenberg type of this matrix defines how the spectral branches merge at the EPs and which types of cycles (rings) they form in the vicinity of the EPs. Once this unfolding mechanism of EPs is generic its fundamental role in many other physical models can be anticipated. A few technical aspects of the used Newton polygon technique are discussed.