Functional Analysis Essentials for Quantum Mechanics

Functional analysis makes limiting arguments part of the specification of a quantum model. To read an operator proof, one needs to distinguish a complete state space, a dense collection of trial vectors, continuity of a map, and closedness of its graph. This page supplies diagnostic calculations for those distinctions. General constructions belong to Hilbert Spaces; quantum-specific domain theorems belong to Operators, Domains, and Spectra.

Helpful background. L2 Spaces supplies equivalence classes of square-integrable functions; Completeness and Orthonormal Bases supplies norm-convergent expansions.

Completeness and the meaning of approximation

A Banach space is a normed vector space in which every Cauchy sequence converges. A Hilbert space is a complete inner-product space. Completeness answers whether a sequence of increasingly accurate approximations has a limit inside the chosen space; it does not answer whether an unbounded observable is defined on that limit.

For example, finite sequences are dense in $\ell^2(\mathbb N)$ but are not complete in the $\ell^2$ norm. Truncating any square-summable sequence produces finite sequences converging to it. Thus finite linear combinations of basis vectors suffice for approximating states, even though restricting all states to such combinations would omit legitimate limits.

A closed subspace $M$ of a Hilbert space admits an orthogonal projection. The requirement that $M$ be closed matters: the dense subspace of finite sequences has orthogonal complement zero, but it is not the whole Hilbert space. Replacing a subspace by its orthogonal complement twice gives its closure, not necessarily the original subspace.

Continuity controls which limits pass through a map

A linear map $B:\mathcal H\to\mathcal K$ is continuous exactly when there is a finite $C$ such that

$$\|B\psi\|\le C\|\psi\|.$$

Consequently $\psi_n\to\psi$ implies $B\psi_n\to B\psi$. This estimate is the reason bounded operators can be extended uniquely from a dense subspace. The extension is obtained by taking limits, and the estimate makes the result independent of the approximating sequence.

The Riesz representation theorem supplies a complementary tool: every continuous linear functional $f$ on a Hilbert space has the form $f(\psi)=\langle\eta,\psi\rangle$ for a unique $\eta$. Continuity is essential. Point evaluation on $L^2(\mathbb R)$ is not such a functional; representatives can differ at one point, and normalized narrow peaks can have arbitrarily large point values. A position ket therefore requires a test-function space, not an additional ordinary Hilbert vector.

Graph norms control the observable as well as the state

For an operator $A$ with domain $D(A)$, define

$$\|\psi\|_A^2=\|\psi\|^2+\|A\psi\|^2.$$

The operator is closed exactly when $D(A)$ is complete in this graph norm. Indeed, a graph-norm Cauchy sequence produces limits $\psi_n\to\psi$ and $A\psi_n\to\eta$ in the Hilbert spaces. Closedness says precisely that $\psi\in D(A)$ and $A\psi=\eta$. Conversely, graph-norm completeness ensures this conclusion whenever both limits exist.

For a dimensional physical observable, insert a fixed positive reference scale in the second term. This gives an equivalent norm and changes none of the topology. The mathematical expression above treats $A$ in fixed units.

Consider the diagonal operator $Ne_j=je_j$ on $\ell^2(\mathbb N)$, with

$$D(N)=\left\{c:\sum_{j=1}^{\infty}j^2|c_j|^2<\infty\right\}.$$

The sequence $\psi_n=e_n/n$ tends to zero in Hilbert norm, but $N\psi_n=e_n$ does not tend to zero in norm. Thus a small state-vector error need not be a small error after applying an observable. In contrast, truncations of a fixed vector in $D(N)$ converge in graph norm because both relevant series tails vanish. This is the practical distinction between a dense approximation set and an operator core.

Three theorems to recognize in a proof

The closed graph theorem says that an everywhere-defined closed linear map between Banach spaces is bounded. It explains why a closed unbounded quantum operator must have a proper domain. It does not say that every everywhere-defined algebraic linear map is continuous.

The uniform boundedness principle says that a family of bounded linear maps which is bounded on every fixed input vector has a uniform operator-norm bound. Its completeness hypothesis prevents pointwise control from hiding arbitrary growth elsewhere.

The bounded inverse theorem says that a bounded bijection between Banach spaces has bounded inverse. For a closed unbounded operator, one instead regards its domain as a Banach space with graph norm before using this theorem. These statements are imported functional-analysis results; Teschl develops their role in the Hilbert-space and operator arguments cited below.

Exercise: identify the missing estimate

Suppose a numerical approximation gives $\|\psi_n-\psi\|\to0$. Explain why probabilities for a fixed bounded effect converge, and why this alone does not control energy expectations for an unbounded Hamiltonian.

Solution

For $0\le E\le I$, subtract the two quadratic expressions and use Cauchy–Schwarz:

$$|\langle\psi_n,E\psi_n\rangle-\langle\psi,E\psi\rangle| \le \|\psi_n-\psi\|(\|\psi_n\|+\|\psi\|).$$

The right side tends to zero. No finite operator-norm bound is available for an unbounded $H$. One needs extra control, such as graph-norm convergence on $D(H)$ or the appropriate form-norm convergence for a semibounded $H$. For a normalized sequence whose mean energy actually diverges despite norm convergence, work the prerequisite map’s second diagnostic.

References

• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — Hilbert-space language and unbounded-operator examples.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition and contents — completeness, bounded maps, and the functional-analytic theorems used here.