Day 247: Spectral Theorem for Compact Self-Adjoint Operators

Learning Objectives

• •**State and prove** the spectral theorem for compact self-adjoint operators
• •**Apply** the Hilbert-Schmidt theorem to characterize eigenvalue sequences
• •**Construct** the spectral decomposition $A = \sum \lambda_n |e_n\rangle\langle e_n|$
• •**Derive** trace formulas and trace-class operator properties
• •**Connect** the spectral decomposition to quantum state expansions
• •**Compute** eigenvalue decompositions numerically and verify convergence

Today's Schedule (7 hours)