Operators, Domains, and Spectra

Infinite-dimensional quantum operators are formula–domain pairs. Formal Hermiticity or real expectation values on a convenient test domain do not establish the self-adjointness needed for a real spectral measure or unitary evolution.

Enter the chapter

On-ramp. State Vectors supplies the Hilbert-space vectors, inner products, and norm convergence used below. The operator treatment also assumes introductory functional analysis: dense subspaces, continuous linear functionals, and closed operators.

Self-Adjoint Operators is the foundation of the rigorous operator sequence. It develops:

• dense domains, adjoints, closedness, and closures;
• the difference between symmetric, self-adjoint, and essentially self-adjoint operators;
• boundary forms, deficiency indices, and the extension criterion;
• real-spectrum and resolvent consequences;
• the links to spectral calculus and Stone’s theorem.

The central identity

$$A=A^*$$

means equality of both action and domain. Every later occurrence of a self-adjoint observable or generator inherits that requirement.

Continue through the spine

From this chapter, two branches open:

• The Unbounded Spectral Theorem constructs spectral measures and functions of $A$.
• Strongly Continuous Unitary Groups prepares the generator theorem for dynamics.

These branches reconverge at Stone’s Theorem. Complete both before that theorem: its forward direction uses spectral calculus to construct $e^{-itA}$, while its converse recovers a generator from the strong derivative domain of the unitary group.

Core Formalism retains the physicist-facing warning about Hermitian versus self-adjoint operators; this chapter owns the operator-theoretic consequences used by the rigorous theorem sequence.

Further topics

Additional domain, extension, form, and spectral-analysis entries are visible with Planned labels and do not yet contain articles. Use the substantive Self-Adjoint Operators treatment above for the definitions and arguments needed by the spectral and dynamical continuations.