Unbounded Operators and Self-Adjointness

The most important operators in physics — position, momentum, energy — are unbounded, defined only on a dense subspace, and this is not a technicality: the distinction between symmetric and self-adjoint decides whether an observable has a spectral theorem and generates a unitary dynamics. This page develops domains, self-adjointness, and self-adjoint extensions. It builds on bounded-operators.md and spectral-theorem.md.

Why unboundedness is forced

The canonical commutation relation $[\hat{x},\hat{p}]=i\hslash$ cannot hold for bounded operators (taking traces or norms gives a contradiction — the Wintner–Wielandt argument). So the operators of quantum mechanics are necessarily unbounded, hence (by the closed graph theorem) not defined on all of $\mathcal{H}$. An unbounded operator $A$ comes with a domain $D(A)⊊\mathcal{H}$, a dense subspace on which it acts; specifying the domain is part of specifying the operator. For $\hat{p}=-i\hslash d/dx$ on $L^2(\mathbb{R})$, $D(\hat{p})$ is (roughly) the differentiable $L^2$ functions with $L^2$ derivative — a Sobolev space, not all of $L^2$.

Symmetric vs. self-adjoint

For a densely defined $A$, the adjoint $A^{†}$ is defined on $$D(A^{†})=\{y:x\mapsto ⟨y,Ax⟩\text{ is bounded on }D(A)\},$$ by $⟨A^{†}y,x⟩=⟨y,Ax⟩$. Two distinct notions now separate that coincide for bounded operators:

• Symmetric (Hermitian): $⟨Ax,y⟩=⟨x,Ay⟩$ for $x,y\in D(A)$ — equivalently $A\subseteq A^{†}$ ($A^{†}$ extends $A$).
• Self-adjoint: $A=A^{†}$, meaning also $D(A)=D(A^{†})$ — the domains match exactly.

The gap $D(A)⊊D(A^{†})$ is invisible in finite dimensions but decisive here:

Only self-adjoint operators have a spectral theorem, real spectrum, and (by Stone's theorem) generate a unitary group $e^{-iAt}$. A merely symmetric operator may have complex spectrum and no well-defined dynamics.

So "the Hamiltonian is Hermitian" is not enough for quantum mechanics to make sense — it must be (essentially) self-adjoint.

No eigenvectors in $\mathcal{H}$

The position and momentum operators have purely continuous spectrum (bounded-operators.md): $\hat{x}|x⟩=x|x⟩$ has no solution $|x⟩\in L^2$ (a delta function is not square- integrable), and $\hat{p}|p⟩=p|p⟩$ gives the non-normalizable plane wave $e^{ipx/\hslash }$. The spectral theorem still applies — $\hat{x}$ is multiplication by $x$ — but the "eigenkets" $|x⟩,|p⟩$ live in the larger rigged Hilbert space, not in $\mathcal{H}$. This is the rigorous meaning of the improper states used throughout QFT/fock-space-inventory.md.

Self-adjoint extensions and deficiency indices

A symmetric operator may admit zero, one, or many self-adjoint extensions, classified by von Neumann:

Deficiency indices $n_{\pm }=\dim \ker (A^{†}\mp i)$ count solutions of $A^{†}\psi =\pm i\psi$. Self-adjoint extensions exist iff $n_{+}=n_{-}$, and are then parameterized by the unitaries $U(n_{+})\to$ (a $U(n_{+})$-family). If $n_{+}=n_{-}>=0$, $A$ is essentially self-adjoint — a unique extension.

Physically the extensions are boundary conditions. Momentum on a finite interval $[0,L]$ has $n_{+}=n_{-}=1$, and the $U(1)$ family of extensions is the choice of phase $\psi (L)=e^{i\theta }\psi (0)$ — periodic, anti-periodic, etc. The same mechanism fixes boundary conditions at singular potentials and the self-adjoint extensions of the Aharonov–Bohm and Calogero problems.

Essential self-adjointness of Hamiltonians

To know a Hamiltonian defines a unique dynamics one proves essential self-adjointness on a convenient domain (e.g. smooth compactly supported functions):

• Kato–Rellich theorem: $H=H_0+V$ is self-adjoint on $D(H_0)$ if $V$ is "$H_0$-bounded" with relative bound $<1$ — covering atomic Hamiltonians (the Coulomb potential is Kato-small relative to $-\Delta$), which is why the hydrogen atom has a well-posed spectral problem.
• Confining or sufficiently regular potentials give essentially self-adjoint $H$ on $C_c^{\infty }$; pathological potentials can fail, signalling that extra physics (boundary data) is needed.

References

• Reed & Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Ch. X.
• Hall, Quantum Theory for Mathematicians, Ch. 9.
• Bonneau, Faraut & Valent, "Self-adjoint extensions of operators and the teaching of quantum mechanics", Am. J. Phys. 69, 322 (2001).