Spectral Analysis of Differential Operators: Worked Solutions to Twenty-Five Problems

Abstract

Study notes with fully worked solutions to twenty-five problems on the spectral analysis of differential and integral operators. The problems cover momentum operators and Laplacians with Dirichlet, Neumann, Robin and periodic boundary conditions; Schrödinger operators with delta, compactly supported and inverse-square potentials; the harmonic oscillator; compact integral and Volterra operators; clamped and Navier bilaplacians; weighted, degenerate and indefinite Sturm–Liouville problems; a one-dimensional Dirac operator; and the unilateral shift. For each operator the solutions establish self-adjointness (via quadratic forms, deficiency indices and von Neumann theory, or Kato–Rellich), compute the spectrum, and classify it into point, continuous, residual, discrete and essential parts. The problems were posed by Niyaz Tokmagambetov (Institute of Mathematics and Mathematical Modeling, Almaty) as part of graduate-level study in functional analysis and spectral theory. Part A collects background definitions and theorems, Part B solves Problems 1–8, and Part C solves Problems 9–25. These are study notes and do not claim new results. An AI assistant (Claude, Anthropic) was used to help draft, translate and typeset the notes; the author is responsible for the content.