Abstract
Some particular properties of the parametric dependence of eigenvalues with emphasis on their complexification are discussed. The non-diagonalisability of PT-symmetric matrix Hamiltonians in exceptional points is compared with level-crossing prohibition of Hermitian systems. For non-matrix Hamiltonians, the different way of complexification between Klein-Gordon and Dirac Hamiltonians is demonstrated.
Highlights
In physics there are usually one or more free parameters which specify some properties of the system
In quantum physics, where almost all physical properties are related to spectral properties of a chosen set of operators, qualitative changes of spectra due to varying parameters of the Hamiltonian are of prominent importance
To show that in less standard situations, there are more ways in which the eigenvalues can cross the boundary between the physical world and the complex realm
Summary
In physics there are usually one or more free parameters which specify some properties of the system. Most attention has been concentrated on discrete spectra, for the following reasons It is usually more convenient (at least for mathematically less careful and less rigorous physicists) to deal with isolated eigenvalues than with the continuous part of the spectrum, where all manipulations become in a way more mathematically intricate. After examining many of the “classical” examples of PT-symmetric systems an apparently regular pattern was observed – a square-root singularity structure and a Jordan-block degeneracy. Though this is beyond the scope of this article, it can be useful for the reader to consider one more application of studying complexification and its properties. In the first section we will make clear the terminology and compare particular properties of self-adjoint and PT-symmetric parametrically dependent Hamiltonians. We will present the situation in relativistic models
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
