Lecture 08 - Spectra and perturbation theory (Schuller's Lectures on Quantum Theory)

Video Lecture:

Summary of contents: resolvent map and resolvent set of an operator; spectrum of an operator; eigenvalues and eigenvectors; difference between spectrum and eigenvalues; pure point spectrum, point embedded in continuum spectrum, purely continuous spectrum; point and continuous spectrum; proof that the eigenvalues of a self-adjoint operator are real; proof that the point spectrum coincides with the eigenvalues; proof by contrapositive; eigenspace and degenerate eigenvalues; proof that the eigenvectors of a self-adjoint operator corresponding to distinct eigenvalues are orthogonal; perturbation theory for the point spectrum; formal power series ansatz; Big O notation; fixing phase and normalisation of perturbed eigenvectors; order-by-order decomposition of the perturbed eigenvalue problem: explicit calculation of first and second-order corrections.

Full lecture notes (work in progress): click here

Lecture Notes for this lecture: