Marko Žnidarič: Pseudospectra

In physics as well as elsewhere one often relies on spectral properties to infer the long-time behavior. For instance, for equilibrium physics at low temperatures it is the energy difference between the ground and the first excited state that matters, for nonequilibrium physics described by Lindblad equation, or Markovian processes, it is the spectral gap. It turns out that when dealing with non-normal linear operators sometimes the spectrum can be irrelevant, even more, it can lead to incorrect conclusions. What can be of use in such situations is a pseudospectrum. I will introduce the concept of a pseudospectrum and show few examples of its use: transfer matrices in 2D quadratic systems, entanglement in random circuits, and certain partial differential equations.