Assess graduate-level understanding of the essential spectrum for unbounded self-adjoint operators and how it behaves under compact and relatively compact perturbations. Items will require precise definitions (essential spectrum, Weyl criterion, relative compactness of operators/resolvents), proofs of invariance results (Weyl's theorem, Kato–Rellich type statements), analysis of concrete examples (Schrödinger and differential operators), and construction of counterexamples when hypotheses fail.