Assess students' ability to apply the spectral theorem and functional calculus for self-adjoint (or nonnegative) operators on Hilbert spaces to construct and analyze solutions of linear evolution equations: derive the heat solution via the contraction semigroup e^{-tA} and the Schrödinger solution via the unitary group e^{-itA}, produce spectral integral representations, verify well-posedness, norm conservation or decay, address domain and essential self-adjointness issues, and work through an explicit example (e.g., Laplacian on a domain or R^n) including point and continuous spectrum considerations.