Spectral Analysis of Differential Operators on Hilbert Spaces

Abstract

Spectral analysis provides a unified framework for investigating differential operators acting on infinite dimensional Hilbert spaces. In contrast with finite-dimensional matrix theory, differential operators are frequently unbounded, and therefore their domains, boundary conditions, closedness, and self-adjoint realizations are essential components of the spectral problem. This paper develops a systematic account of the spectrum, resolvent, point and continuous spectral components, compact resolvents, spectral measures, variational principles, and spectral approximation for self-adjoint differential operators. Particular attention is given to Sturm–Liouville operators, Dirichlet and Neumann Laplacians, and one dimensional Schrödinger operators. The role of boundary conditions in changing the spectrum is made explicit, and the distinction between discrete and essential spectra is emphasized. The Rayleigh quotient and min–max principle are used to connect operator theory with energy methods and numerical eigenvalue approximation. Finite-difference
and Galerkin discretizations are then developed, including residual-based diagnostics, convergence under mesh refinement, and the phenomenon of spectral pollution. For the Dirichlet Laplacian on (0,1), the discrete eigenvalues are obtained in closed form, and their second-order convergence to the exact spectrum is derived analytically and confirmed numerically. The discussion is further extended to perturbation of
self-adjoint operators, Weyl-type stability of the essential spectrum, semigroup representations for evolution equations, and applications in vibration, diffusion, quantum mechanics, stability analysis, and inverse
problems. The resulting framework highlights the central role of self-adjointness, compactness, variational structure, and structure-preserving discretization in obtaining mathematically reliable and computationally
stable spectral information.