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.

From operators to observables and dynamics

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.

For a first boundary-condition example, see Hermitian versus Self-Adjoint Operators. Return here for the domain and extension arguments behind that example.

Build the operator language

Begin with Operators in Quantum Mechanics to separate transformations, observables, events, and generators. Then compare Bounded Operators with Unbounded Operators: the latter require domains because norm convergence of inputs need not control their images.

The next sequence supplies the tools for proving operator statements:

• Dense Domains distinguishes ordinary density, graph cores, form cores, and invariant test spaces.
• Closed and Closable Operators explains which simultaneous limits define a stable action.
• Adjoints of Unbounded Operators computes maximal domains from weak derivatives and boundary pairings.
• Symmetric Operators identifies what real expectations and nonreal range estimates establish before self-adjointness is proved.

Choose a realization and analyze its spectrum

Essential Self-Adjointness answers when a small test domain determines a unique self-adjoint closure. When a choice remains, Self-Adjoint Extensions constructs the larger domains, and Deficiency Indices provides explicit existence and dimension calculations. Quadratic Forms gives a complementary construction that works with finite-energy vectors and weak equations.

For spectral calculations, proceed through Spectra of Operators, Point, Continuous, and Residual Spectrum, and Resolvent Operators. Functional Calculus then explains how powers, phases, inverses, and cutoffs inherit their domains from spectral integrability. Use Common Operator Pitfalls to diagnose a calculation whose formal algebra and domain assumptions disagree.

References

• B. C. Hall, Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — the connection between physical operator language and functional analysis.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — a proof-oriented route from self-adjointness to quantum dynamics.