Bounded and Unbounded Operators on Hilbert Space

Self-adjoint extension theory is not a museum piece; it is the working language of several live areas of physics. In cold-atom laboratories a Bose–Einstein condensate confined to a ring and threaded by a synthetic gauge field realizes the $\theta$-family of extensions directly — the boundary phase $\theta$ is a tunable magnetic flux, and the persistent currents it drives are the physical readout of which self-adjoint operator nature has selected, exactly the $p_n=\hbar(2\pi n+\theta)/L$ ladder of the second derivation. In few-body physics, point (contact) interactions and the attractive $1/r^2$ potential are formally symmetric but not self-adjoint; choosing a self-adjoint extension is a genuine physical input, and it is the mechanism behind the Efimov effect and the log-periodic bound-state towers seen in ultracold trimers. At the foundational end, the Kato–Rellich theorem is what guarantees that the molecular Hamiltonians solved every day in computational quantum chemistry are essentially self-adjoint, so their computed spectra are meaningful rather than artifacts of a truncation. And the frontier remains open: proving essential self-adjointness of interacting quantum-field Hamiltonians, and understanding self-adjoint extensions on quantum graphs and near black-hole singularities, are active research problems where the century-old deficiency-index bookkeeping is still the decisive tool.

Think of an operator's norm as its worst-case stretch factor, taken over every unit vector in the space. In finite dimensions there are only finitely many independent directions, so the largest stretch is achieved and finite — every matrix is bounded, defined everywhere, automatically continuous. In infinite dimensions there are infinitely many mutually orthogonal directions, and an operator can stretch each successive direction harder than the last with no ceiling. Differentiation does precisely this: the orthonormal Fourier modes $e_n(x)=e^{inx}/\sqrt{2\pi}$ each have unit norm, yet $\mathrm{d}e_n/\mathrm{d}x=in\,e_n$ has norm $n$. There is no finite $C$ with $\|Ax\|\le C\|x\|$ for all $x$, so the derivative simply cannot be a continuous operator on the whole space. The first figure makes this concrete: the amplification climbs as $1,2,3,\dots$ and never levels off, in contrast to a bounded operator whose stretch is capped.

A self-adjoint operator is the rigorous version of "an observable with real values and a complete set of states." The subtlety unique to infinite dimensions is that completeness can be continuous. Position has no normalizable eigenstates at all, yet $\int|x\rangle\langle x|\,\mathrm{d}x=\mathbf{1}$ in the distributional sense, and the spectral theorem replaces the sum over eigenprojectors by an integral over a projection-valued measure. Diagonalization survives even when there are literally no eigenvectors in the space. But this only works when the operator is self-adjoint, not merely symmetric — and whether it is depends on the domain, which is why domains carry physics. The second figure shows the payoff: the spectrum of a self-adjoint operator is pinned to the real axis, and every complex energy is separated from it by a resolvent bound that a symmetric-but-not-self-adjoint operator fails to satisfy.

Boundedness and self-adjointness are independent axes, and all four corners are physically occupied. A projection is bounded and self-adjoint; a generic compact operator is bounded but not self-adjoint; the Hamiltonian is unbounded and self-adjoint; momentum on a finite interval is unbounded and merely symmetric. Keeping the two properties apart — one about continuity, one about spectral theory — is the whole conceptual content of the subject.

The operator norm is the single number that decides continuity. It is the supremum of the amplification factor $\|Ax\|/\|x\|$ over all nonzero vectors, equivalently the largest norm $A$ produces on the unit sphere. When this supremum is finite the operator is bounded; when it is infinite the operator is unbounded and cannot live on the whole space.

where: $A$ — linear operator on the Hilbert space $\mathcal{H}$; $\|x\|=\sqrt{\langle x,x\rangle}$ — the Hilbert-space norm; $\|A\|$ — operator (supremum) norm, finite if and only if $A$ is bounded. Reading: $\|A\|$ is the worst-case stretch; if it is finite the operator is Lipschitz, hence continuous everywhere. The adjoint generalizes the conjugate transpose. It is defined by the requirement that moving $A$ from the right slot of the inner product to the left slot produces $A^\dagger$, for every vector in the domain of $A$. For bounded operators this defines $A^\dagger$ on all of $\mathcal{H}$; for unbounded operators the adjoint carries its own, generally different, domain.

where: $A^\dagger$ — the adjoint of $A$; $D(A)$ — the dense domain of $A$; $\langle\cdot,\cdot\rangle$ — the inner product, taken antilinear in the first slot. Reading: the adjoint is whatever operator makes this identity hold; existence of $A^\dagger y$ requires the map $x\mapsto\langle y,Ax\rangle$ to be a bounded functional. That last requirement is the domain of the adjoint. A vector $y$ belongs to $D(A^\dagger)$ precisely when the functional $x\mapsto\langle y,Ax\rangle$ is bounded on $D(A)$, so that the Riesz representation theorem supplies a unique vector to call $A^\dagger y$.

where: $y$ — a candidate vector for the adjoint domain; the functional must extend from the dense $D(A)$ to a bounded functional on all of $\mathcal{H}$. Reading: shrinking $D(A)$ enlarges $D(A^\dagger)$, because fewer test vectors $x$ make the boundedness condition harder to violate. Self-adjointness is the equality of an operator with its adjoint as operators — same action and same domain. This is strictly stronger than symmetry, in which $A^\dagger$ merely extends $A$ to a possibly larger domain. The whole difficulty of unbounded operators is compressed into the gap between the two.

where: $A=A^\dagger$ — action and domain both coincide; $A\subseteq A^\dagger$ — symmetric, meaning $A^\dagger$ agrees with $A$ on $D(A)$ but may act on a strictly larger domain. Reading: symmetry is a formal computation with the inner product; self-adjointness is an assertion about domains that must be checked separately. The deficiency indices measure exactly that gap. They count the square-integrable solutions of $A^\dagger\psi=\pm i\psi$ — the eigenvectors of the adjoint at the two non-real eigenvalues $\pm i$, which can only exist because $A^\dagger$ extends beyond the symmetric $A$.

where: $n_+,n_-$ — the deficiency indices. Reading: $n_+=n_-=0$ means $A$ is essentially self-adjoint (unique good extension); $n_+=n_-=n\ge 1$ gives a $U(n)$ family of self-adjoint extensions; $n_+\neq n_-$ means no self-adjoint extension exists at all. The spectral theorem is the reward. For a self-adjoint operator it writes $A$ as an integral of the spectral parameter $\lambda$ against a projection-valued measure $E_\lambda$, the resolution of the identity — the continuous replacement for a sum over eigenprojectors.

where: $\sigma(A)\subseteq\mathbb{R}$ — the spectrum; $E_\lambda$ — the projection-valued spectral measure; $\mathrm{d}E_\lambda$ — the orthogonal projection onto the spectral subspace near $\lambda$. Reading: functions of $A$ are defined by $f(A)=\int f(\lambda)\,\mathrm{d}E_\lambda$, which is how $e^{-iAt}$ acquires meaning. The spectrum itself generalizes the eigenvalue set. It is the set of complex $\lambda$ for which $A-\lambda$ fails to have a bounded inverse, and it splits into three disjoint pieces according to how the inverse fails.

where: $\sigma_p$ — point spectrum, $A-\lambda$ not injective (true eigenvalues); $\sigma_c$ — continuous spectrum, injective with dense but not closed range (no eigenvector); $\sigma_r$ — residual spectrum, range not dense. Reading: for self-adjoint $A$ the residual part is empty and the whole spectrum is real. Stone's theorem ties self-adjointness to dynamics. It is a bijection: every self-adjoint generator produces exactly one strongly continuous one-parameter group of unitaries, and every such group has exactly one self-adjoint generator.

where: $H$ — the self-adjoint generator (Hamiltonian); $U(t)$ — the evolution, with $U(t)U(s)=U(t+s)$, $U(0)=\mathbf{1}$, $U(t)^\dagger=U(t)^{-1}$; $\hbar$ — the reduced Planck constant. Reading: unitarity is what conserves probability, and it is available only for a self-adjoint $H$, not a merely symmetric one. Finally, the canonical commutation relation shows why the players are unbounded in the first place. Position and momentum cannot both be bounded and satisfy it — the Wintner–Wielandt argument shows $[\hat x,\hat p]=i\hbar\mathbf{1}$ is impossible for bounded operators — so at least one must be unbounded, defined only on a common invariant core.

where: $\hat x,\hat p$ — position and momentum, both unbounded self-adjoint on dense domains; the relation holds on a common core such as Schwartz space $\mathcal{S}(\mathbb{R})$. Reading: the commutator forces unboundedness, which is why the domain apparatus is unavoidable in quantum mechanics.

Where the infinite-dimensional theory collapses back onto familiar finite-dimensional facts, and where it genuinely departs from them:

Example 1 — the operator norm of a bounded multiplication operator

Let $(M\psi)(x)=x\,\psi(x)$ act on $L^2([0,1])$. Show it is bounded, find its operator norm, and confirm it is self-adjoint. Then contrast with the same symbol on $L^2([0,\infty))$.

Step 1. Estimate the stretch. For any $\psi$, $\|M\psi\|^2=\int_0^1 x^2|\psi(x)|^2\,\mathrm{d}x\le\int_0^1 1\cdot|\psi(x)|^2\,\mathrm{d}x=\|\psi\|^2$, since $x\le 1$ on $[0,1]$. Hence $\|M\psi\|\le\|\psi\|$ and $M$ is bounded with $\|M\|\le 1$.

Step 2. Show the bound is sharp. Concentrate the mass near $x=1$: take $\psi_\epsilon$ normalized and supported on $[1-\epsilon,1]$. Then $\|M\psi_\epsilon\|^2=\int_{1-\epsilon}^1 x^2|\psi_\epsilon|^2\ge(1-\epsilon)^2$, so $\|M\psi_\epsilon\|\to 1$ as $\epsilon\to 0$. The supremum is $\|M\|=1$, but it is not attained — there is no unit vector with $\|M\psi\|=1$ exactly, a purely infinite-dimensional phenomenon. The operator norm is $\|M\|=1$, an unattained supremum.

Step 3. Check self-adjointness. Since $x$ is real, $\langle M\phi,\psi\rangle=\int_0^1\overline{x\phi}\,\psi=\int_0^1\overline{\phi}\,x\psi=\langle\phi,M\psi\rangle$, and because $M$ is bounded its domain is all of $L^2([0,1])=D(M^\dagger)$. So $M=M^\dagger$: bounded and self-adjoint. Its spectrum is the range of the multiplier, $\sigma(M)=[0,1]$, which is purely continuous — no eigenvectors, since $x\psi=\lambda\psi$ forces $\psi=0$ almost everywhere.

Step 4. Contrast on the half-line. On $L^2([0,\infty))$ the same symbol has $\|M\psi\|^2=\int_0^\infty x^2|\psi|^2$, which is unbounded: a unit bump near $x=R$ gives $\|M\psi\|\approx R\to\infty$. The identical formula is bounded on $[0,1]$ and unbounded on $[0,\infty)$ — boundedness is a property of the operator-plus-domain, never of the symbol alone.

Example 2 — one differential symbol, two spectra (fully worked with numbers)

The formal operator is $\hat H=-\dfrac{\hbar^2}{2m}\dfrac{\mathrm{d}^2}{\mathrm{d}x^2}$ on an interval of length $L$. Compute the energy spectrum for two self-adjoint extensions — the Dirichlet box and the periodic ring — for an electron with $L=1\ \mathrm{nm}$, and see how the choice of domain changes the physics.

Step 1. Dirichlet box. The domain requires $\psi(0)=\psi(L)=0$; eigenfunctions $\sin(k_nx)$ force $k_n=n\pi/L$ with $n=1,2,3,\dots$, giving

Step 2. Put in numbers with $\hbar=1.055\times10^{-34}\ \mathrm{J\,s}$, $m=9.109\times10^{-31}\ \mathrm{kg}$, $L=10^{-9}\ \mathrm{m}$. The ground state is $E_1^{\text{box}}=\pi^2\hbar^2/(2mL^2)=6.02\times10^{-20}\ \mathrm{J}=\mathbf{0.376\ eV}$, and the levels run as $n^2$: $E_1:E_2:E_3=0.376:1.504:3.384\ \mathrm{eV}$. There is no $n=0$ state — a zero function is not a state.

Step 3. Periodic ring. The domain requires $\psi(L)=\psi(0)$ (the $\theta=0$ extension); eigenfunctions $e^{ik_nx}$ force $k_nL=2\pi n$, so $k_n=2\pi n/L$ with $n\in\mathbb{Z}$ (including $0$), giving

Step 4. Evaluate. Now $E_0^{\text{ring}}=0$ (a legitimate constant state, unlike the box), and $E_1^{\text{ring}}=2\pi^2\hbar^2/(mL^2)=2.41\times10^{-19}\ \mathrm{J}=\mathbf{1.504\ eV}$, each level $n\neq 0$ doubly degenerate (clockwise and anticlockwise). Note the arithmetic coincidence $E_1^{\text{ring}}=E_2^{\text{box}}=1.504\ \mathrm{eV}$ — the same number, reached through two different quantizations.

Step 5. Read the physics. The symbol $-\hbar^2/2m\,\mathrm{d}^2/\mathrm{d}x^2$ is identical in both cases; only the domain differs, yet the spectra differ qualitatively — a nonzero ground state and no degeneracy in the box, a zero-energy state and twofold degeneracy on the ring. Same operator symbol, two self-adjoint extensions, two physically distinct energy spectra — the concrete meaning of "the domain is part of the operator."

The rigorous theory of unbounded operators has two founders working in parallel around 1930 — von Neumann, who defined self-adjointness and the deficiency indices, and Marshall Stone, whose theorem on unitary groups turned the abstract spectral theorem into the engine of quantum dynamics.

Texts. Stone & Goldbart, Mathematics for Physics (Cambridge, 2009), Ch. 2 & 9 · Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 5 & 8 · Hassani, Mathematical Physics, 2nd ed., Ch. 17 (unbounded operators) · Reed & Simon, Methods of Modern Mathematical Physics I & II (Functional Analysis; Self-Adjointness) · von Neumann, Mathematical Foundations of Quantum Mechanics (1932/1955) · Byron & Fuller, Mathematics of Classical and Quantum Physics, Ch. 5.

Provenance. A verifier should confirm: (1) the operator-norm/continuity equivalence and the Lipschitz bound $\|Ax-Ay\|\le\|A\|\,\|x-y\|$ (Reed & Simon I; Stone & Goldbart Ch. 2); (2) the derivative example $\|Ae_n\|=n$ on $L^2([0,2\pi])$; (3) the integration-by-parts boundary term and the deficiency indices $(1,1)$ for momentum on $[0,L]$, with extensions $\psi(L)=e^{i\theta}\psi(0)$ and spectrum $p_n=\hbar(2\pi n+\theta)/L$ (von Neumann; Reed & Simon II); (4) the resolvent bound $\|(A-\lambda)^{-1}\|\le 1/|\mathrm{Im}\,\lambda|$ proving $\sigma(A)\subseteq\mathbb{R}$ (Reed & Simon I, Thm VIII.3); (5) the box and ring energies $E_1^{\text{box}}=0.376\ \mathrm{eV}$ and $E_1^{\text{ring}}=1.504\ \mathrm{eV}$ for an electron with $L=1\ \mathrm{nm}$. All indices and numbers are standard textbook values.

Prerequisites. Inner-Product Spaces and Norms · Adjoint Operators and Hermitian Operators · Linear Operators and Matrix Representations · Orthogonality and the Gram–Schmidt Process. Leads to. Diagonalization and the Spectral Decomposition · Eigenvalues, Eigenvectors, and the Characteristic Equation.

Ask the tutor: Ask me to walk you through the deficiency-index computation for any operator you like — momentum on a half-line, the radial kinetic term, or a delta-potential — and we will find its self-adjoint extensions together.