Spectrum of Self-adjoint Operators

This chapter contains multiple tools for studying the spectrum of a self-adjoint operator. First, we discuss how the spectrum is modified when a small perturbation is added to a given operator. Next, we define the different types of spectrum (point, continuous, essential, discrete) and present criteria due to Weyl in order to identify them. Compact and compact-resolvent operators are treated as examples. The Courant-Fischer formula and the Lieb-Thirring inequality are then used to quantify the number and size of eigenvalues below the essential spectrum. The chapter concludes with a short introduction to semi-classical analysis.