Lectures on Quantum Theory (Frederic Schuller)
by Frederic P. Schuller · Friedrich-Alexander-Universität Erlangen-Nürnberg (YouTube)
Recorded lecture course at FAU Erlangen-Nürnberg that builds the operator theory behind quantum mechanics from definitions: Banach and separable Hilbert spaces, closable and adjoint operators, self-adjoint and essentially self-adjoint operators, spectra, perturbation theory and the spectral theorem.
More resources on Linear Operators
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Introduction to Functional Analysis (MIT 18.102)
Analysis on infinite-dimensional spaces: normed and Banach spaces, the Hahn-Banach theorem and duality, bounded operators, Lebesgue measure and Lp spaces, Hilbert spaces, compact and self-adjoint operators, and the spectral theorem. Includes 23 lecture videos, extensive lecture notes, problem sets and exams.
Perturbation Theory for Linear Operators
Kato's reference monograph on linear operators in finite-dimensional, Banach and Hilbert spaces. Covers closed and closable operators, adjoints, resolvents, analytic and asymptotic perturbation of eigenvalues, sectorial forms, semigroups and scattering theory. Capstone reference; the only one treating closed operators in Banach spaces and eigenvalue perturbation this exhaustively.
Linear Operators in Hilbert Spaces
Graduate text (Springer GTM 68) on the theory of bounded and unbounded operators in Hilbert spaces, covering self-adjoint extensions, spectral theory, and perturbation theory, with applications to differential operators from quantum mechanics. Assumes measure theory and basic functional analysis.