Nonlinear Spectral Problems

Chapter 10 develops a computational framework for nonlinear spectral problems defined by operator families. Assuming only gap-topology continuity, it extends Chapter 3 to compute spectra and pseudospectra without restricting to discrete spectrum, while controlling spectral pollution and invisibility. SCI classifications show these procedures are optimal, illustrated by acoustic boundary conditions, Orr–Sommerfeld stability, and delay equations. We then treat holomorphic families on regions of discrete spectrum and prove a full-strength Keldysh theorem (allowing unbounded operators and z-dependent domains) linking eigenvalues, multiplicities, and Jordan chains to the Laurent expansion of the inverse family. This enables reductions to finite-dimensional linear eigenproblems via contour integration and numerical quadrature. Finally, a unified analysis of contour-integral methods (encompassing most algorithms in the literature) covers stability, convergence and randomised sketching with probe vectors, yielding parallelisable routines for eigenvalues and generalised eigenvectors. Exercises treat essential spectra, Newton refinements and randomised trace estimates of multiplicities.