Spectral Types of Self-Adjoint and Unitary Operators

Chapter 6 analyses spectral types of self-adjoint and unitary operators—absolutely continuous, singular continuous, and pure point—and the resulting block-diagonal decomposition and Lebesgue decomposition of spectral measures. These spectral types are fundamental to understanding the physical and dynamical behaviour of many operators. Within the Solvability Complexity Index (SCI) hierarchy, we classify the complexity of computing (i) the measure components and (ii) the spectral sets on the line or unit circle. For operator classes whose resolvents can be computed with asymptotic error control, we design optimal two-limit algorithms: RAGE-type dynamical limits separate continuous from pure point parts, while resolvent smoothing combined with Radon–Nikodym estimates extracts the absolutely continuous part. Spectral sets are computed via a dyadic decision tree that tests interval intersections. Sharp lower bounds (impossibility results) are proved for discrete Schrödinger and CMV classes using Anderson localisation and sparse potentials. Worked examples include the unitary almost Mathieu phase transition and extracting eigenvalues embedded in the continuous spectrum.