Spectral Analysis of Dirac Operators with a Purely Imaginary Dislocation

Explore a comprehensive mathematical lecture on the spectral analysis of Dirac operators with purely imaginary dislocations. Delve into the complete spectral analysis of Dirac operators in one dimension with non-Hermitian matrix potentials of the form isgn(x) +V(x) where V ∈ L1. Learn how to explicitly compute the matrix Green function for V = 0, enabling the determination of the purely essential spectrum and its various types. Discover sharp enclosures for the pseudospectrum and its complement across the complex plane, including the instability region surrounding the real axis. Examine how the spectrum and pseudospectrum change when V ≠ 0 under specific hypotheses, and investigate the preservation of essential spectra and behavior of ε-pseudospectrum. Gain insights into sharp asymptotic estimates for the discrete spectrum under certain decay conditions. Conclude with a thorough description of the weakly-coupled model, based on collaborative research with Tho Nguyen Duc and David Krejčiřík from the Czech Technical University in Prague.