Assess students' ability to analyze spectral properties of self-adjoint Schrödinger operators H = −Δ + V on R^n with potentials that decay at infinity and/or have local singularities: test proving self‑adjointness and characterizing quadratic form domains, classifying essential vs. discrete spectrum, deriving eigenvalue accumulation criteria at thresholds (0 or continuum edges) and at infinity, applying Birman–Schwinger and Weyl asymptotic methods, using Hardy inequalities, Mourre or limiting absorption techniques to control embedded eigenvalues and singular continuous spectrum, and estimating eigenvalue counting functions with concrete examples and counterexamples.