Abstract
A variety of physical phenomena, such as amplification, absorption, and radiation, can be effectively described using non-Hermitian operators. However, the introduction of non-uniform non-Hermiticity can lead to the formation of exceptional points in a system’s spectrum, where two or more eigenvalues become degenerate and their associated eigenvectors coallesce causing the underlying operator or matrix to become defective. Here, we explore extensions of the Clifford and quadratic ϵ-pseudospectrum, previously defined for Hermitian operators, to accommodate non-Hermitian operators and matrices, including the possibility that the underlying operators may possess exceptional points in their spectra. In particular, we provide a framework for finding approximate joint eigenstates of a d-tuple of Hermitian operators A and non-Hermitian operators B, and show that their Clifford and quadratic ϵ-pseudospectra are still well-defined despite any non-normality. Altogether, this framework enables the exploration of non-Hermitian physical systems’ ϵ-pseudospectra, including but not limited to photonic systems where gain, loss, and radiation are prominent physical phenomena. This is a story about commuting and non-commuting matrices and their implications. Because of this the only real background knowledge coming in is a good grasp of linear algebra. Knowledge on ϵ pseudospectrum is not required in any form and the goal of the talk will be to introduce key ideas along the way. We will provide a connection between joint approximate eigenstates and the pseudospectrum of some composite operators. We also establish a relationship between the different composite operators and end with some numerical simulations on
non-Hermitian toy model physical systems.

Talk will be in person, but we will set up a zoom link soon.