Assess graduate students' ability to analyze unbounded self-adjoint operators on Hilbert spaces: state and apply the spectral theorem via projection-valued (spectral) measures, construct and compute the Borel functional calculus, address domain and essential self-adjointness issues (deficiency indices, cores), and apply these tools to Schrödinger operators to determine spectrum, spectral types, resolvent behavior, and unitary evolution e^{-itH}; include theorem-proof and problem-solving tasks that require rigorous spectral decompositions and functional-calculus computations.