Unbounded Operators and the Spectral Theorem

The operators that matter most in analysis and geometry are unbounded. Differentiation is unbounded on every $L^p$ space (4A.1 Banach Spaces and Bounded Operators); the Laplacian, the Schrödinger operators of quantum mechanics $-\Delta + V$, and Perelman's operator $-4\Delta + R$ (12A.2 Ricci Flow as a Gradient Flow) are all defined only on a dense subspace of $L^2$, and on that subspace they can make norms arbitrarily large. Yet we want to treat them like symmetric matrices: diagonalise them, take their square roots and exponentials, solve $\partial_tu = \Delta u$ by $u = e^{t\Delta}u_0$ and $i\partial_t\psi = H\psi$ by $\psi = e^{-itH}\psi_0$.

This works, but only for a carefully defined class: the self-adjoint operators. The distinction between "symmetric" ($\langle Au, v\rangle = \langle u, Av\rangle$ on the domain) and "self-adjoint" (symmetric with the right domain) is invisible for matrices and decisive for differential operators. It is where boundary conditions live: $-\frac{d^2}{dx^2}$ on an interval is symmetric on smooth functions vanishing near the ends, and it has many self-adjoint extensions, one for each physically meaningful boundary condition, each with its own spectrum. This chapter develops the theory needed to make these statements precise, and proves the spectral theorem in the form in which the rest of the guide uses it.

By the end of this chapter you will be able to:

• work with closed and closable operators and with adjoints of densely defined operators, and compute the closure of a differential operator;
• distinguish symmetric from self-adjoint operators, and prove the basic criterion $\operatorname{Ran}(A \pm i) = H$;
• construct the Friedrichs extension of a semibounded symmetric operator, and identify it for the Laplacian as the Dirichlet Laplacian;
• prove the spectral theorem for self-adjoint operators with compact resolvent, define the functional calculus $f(A)$, and use it for $e^{-tA}$, $e^{-itA}$ and $A^s$; state the general spectral theorem and Stone's theorem;
• prove the Kato–Rellich theorem and use it to show that atomic Hamiltonians are self-adjoint;
• prove the min–max characterisation of eigenvalues, and apply all of this to $-4\Delta + R$ on a closed manifold.

13.1Observables and boundary conditions

13.2Unbounded operators, graphs and closures

An operator $A$ on a Hilbert space $H$ is a linear map $A : D(A) \to H$ from a subspace $D(A)$, its domain; it is densely defined if $D(A)$ is dense. Two operators are equal only if their domains are; $B$ extends $A$, written $A \subseteq B$, if $D(A) \subseteq D(B)$ and $B = A$ on $D(A)$. The graph $\Gamma(A) = \{(u, Au) : u \in D(A)\}$ is a subspace of $H\times H$.

By the closed graph theorem (4A.2 Baire Category and Its Consequences), a closed operator defined on all of $H$ is bounded. So unbounded closed operators are necessarily defined only on a proper subspace: the domain is part of the operator.

The adjoint. For densely defined $A$, let $D(A^*)$ be the set of $v \in H$ for which $u \mapsto \langle Au, v\rangle$ is bounded on $D(A)$; by Riesz (4A.5 Hilbert Spaces and Lax–Milgram) there is then a unique $A^*v$ with $\langle Au, v\rangle = \langle u, A^*v\rangle$ for all $u \in D(A)$. Density of $D(A)$ makes $A^*v$ unique. Basic facts (Exercise 13.15):

• $A^*$ is always closed: its graph $\{(v, w) : \langle Au, v\rangle = \langle u, w\rangle\ \forall u \in D(A)\}$ is the orthogonal complement $(J\Gamma(A))^\perp$, with $J(x, y) = (-y, x)$.
• $A$ is closable exactly when $A^*$ is densely defined, and then $\bar A = A^{**}$.
• $A \subseteq B$ implies $B^* \subseteq A^*$: extending an operator shrinks its adjoint.

13.3Symmetric and self-adjoint

For a symmetric $A$, $A \subseteq \bar A = A^{**} \subseteq A^*$. Extending a symmetric operator enlarges its domain and shrinks its adjoint's; self-adjointness is the moment the two meet. A self-adjoint operator has no proper symmetric extension ($A \subseteq B \subseteq B^* \subseteq A^* = A$), so self-adjoint operators are maximal symmetric. An essentially self-adjoint operator has exactly one self-adjoint extension, its closure.

The physical meaning: momentum generates translations, and a particle on a half-line cannot be translated past the wall without losing probability, so there is no unitary translation group and no self-adjoint momentum.

Von Neumann's theory. For a closed symmetric $A$, the deficiency spaces $K_\pm = \ker(A^* \mp i)$ have dimensions $n_\pm$, the deficiency indices. Von Neumann proved (1929): $A$ is self-adjoint iff $n_+ = n_- = 0$; $A$ has self-adjoint extensions iff $n_+ = n_-$, and they correspond to unitary maps $K_+ \to K_-$. For the momentum on $(0, 1)$, $n_\pm = 1$ and the extensions are the circle of $P_\alpha$; on the half-line $n_+ = 1 \neq n_- = 0$.

The case $n_\pm = 0$ is the one used in practice, and it has a direct proof.

13.4The Friedrichs extension

For differential operators, the most useful way to produce a self-adjoint operator is through its quadratic form, where Lax–Milgram (4A.5 Hilbert Spaces and Lax–Milgram) does the work.

13.5The spectral theorem

With compact resolvent. Most operators in this guide have compact resolvent: $(A - z)^{-1}$ is compact for some, hence every, $z$ in the resolvent set. This happens for the Dirichlet Laplacian on a bounded domain (Rellich's theorem makes $H^1_0 \hookrightarrow L^2$ compact, 4A.11 Sobolev Embeddings and Critical Exponents) and for $-\Delta + V$ on a closed manifold. Then the spectral theorem for compact self-adjoint operators (4A.8 Compact Operators and Spectra) does everything.

Functional calculus. For any Borel function $f$ on the spectrum, define

Then $f \mapsto f(A)$ is an algebra homomorphism, $\|f(A)\| = \sup_k|f(\lambda_k)|$ for bounded $f$, $\bar f(A) = f(A)^*$, and real $f$ give self-adjoint operators. Three choices of $f$ are used constantly.

• $f(\lambda) = e^{-t\lambda}$: the heat semigroup $e^{-tA}$, $t \geq 0$. For $A = -\Delta_D$, $u(t) = e^{t\Delta}u_0$ solves the heat equation with Dirichlet conditions: $u(t) = \sum e^{-t\lambda_k}\langle u_0, e_k\rangle e_k$, Fourier's own solution (2B.7 Fourier Series and the First Heat Equation), now justified for every $u_0 \in L^2$. It is a contraction, $\|e^{t\Delta}\| = e^{-t\lambda_1} \leq 1$, and smoothing: $\|Ae^{-tA}\| = \sup_k\lambda_ke^{-t\lambda_k} \leq \frac{1}{et}$ for $A \geq 0$, which says that after any positive time the solution is in the domain of every power of $A$, hence smooth by elliptic regularity.
• $f(\lambda) = e^{-it\lambda}$: the unitary group $e^{-itA}$, solving the Schrödinger equation $i\partial_t\psi = A\psi$. It preserves norms and does not smooth.
• $f(\lambda) = \lambda^s$ for $A \geq 0$: fractional powers, such as $(-\Delta)^{1/2}$, the Dirichlet-to-Neumann type operator of 4A.12 Interpolation and Fractional Sobolev Spaces, and $(-\Delta)^{-1}$, the solution operator of the Poisson equation.

The general spectral theorem. Without compact resolvent, the spectrum can be continuous (for $-\Delta$ on $\mathbb{R}^n$ it is $[0, \infty)$ with no eigenvalues at all), and sums become integrals.

For $A = -\Delta$ on $L^2(\mathbb{R}^n)$, $U$ is the Fourier transform and $a(\xi) = 4\pi^2|\xi|^2$ (4A.6 The Fourier Transform): the Fourier transform diagonalises the Laplacian, and the functional calculus is the Fourier multiplier calculus of 4A.12 Interpolation and Fractional Sobolev Spaces. In general $f(A) = U^{-1}f(a)U$. The equivalent "projection-valued measure" form writes $A = \int\lambda\,dE(\lambda)$.

The forward direction is immediate from the multiplication form and dominated convergence (Exercise 13.19). Stone's theorem is why observables generate symmetries in quantum mechanics: momentum generates translations, angular momentum rotations, energy time evolution; and why the momentum on a half-line, which cannot generate translations, cannot be self-adjoint.

13.6Perturbations: Kato–Rellich

To prove self-adjointness of $-\Delta + V$, one perturbs the known self-adjoint $-\Delta$.

13.7Min–max

For self-adjoint operators bounded below with compact resolvent, eigenvalues are values of a variational problem, which is how they are estimated in practice.

Consequences: eigenvalues increase when the form increases (larger potential, larger domain constraints), and Dirichlet eigenvalues decrease when the domain grows ($H^1_0(\Omega) \subseteq H^1_0(\Omega')$ for $\Omega \subseteq \Omega'$), a monotonicity used to prove Weyl's law by bracketing (6B.15 Spectral Theory and Heat Trace Asymptotics). With $k = 1$ it is the Rayleigh quotient of 4A.8 Compact Operators and Spectra and 6B.16 Eigenvalue Problems and Perelman’s λ.

13.8The operator $-4\Delta + R$

On a closed Riemannian manifold $(M, g)$, $L = -4\Delta + R$ with $R$ smooth is the operator whose lowest eigenvalue is Perelman's $\lambda(g)$ (12A.2 Ricci Flow as a Gradient Flow). The theory above gives, with elliptic regularity on manifolds (9B.7 The Heat Equation on a Manifold):

1. $-\Delta$ is essentially self-adjoint on $C^\infty(M)$ (no boundary, so no boundary conditions to choose), with self-adjoint closure on $H^2(M)$ and compact resolvent by Rellich on $M$.
2. Multiplication by $R$ is bounded, so $L$ is self-adjoint on $H^2(M)$ by Kato–Rellich (with $a = 0$), bounded below by $\min R$, with compact resolvent and an orthonormal eigenbasis.
3. $\lambda(g) = \lambda_1(L) = \min\frac{\int(4|\nabla w|^2 + Rw^2)}{\int w^2}$ by min–max, and the minimiser can be taken positive: if $w$ minimises, so does $|w|$ (with $|\nabla|w|| = |\nabla w|$ almost everywhere), which then solves $Lw = \lambda_1w$, is smooth, and by the strong maximum principle (6A.5 Maximum Principles) is everywhere positive. Two positive eigenfunctions cannot be orthogonal, so $\lambda_1$ is simple.
4. The heat semigroup $e^{-tL}$ and its kernel exist, and $e^{-tL}$ maps positive functions to positive functions (it is positivity preserving, by the maximum principle); this is the operator-theoretic home of the conjugate heat equation $\square^*u = 0$, whose solutions Perelman runs backward in time (12A.3 The 𝓦-Entropy).

Simplicity and positivity of the ground state are what make $\lambda(g)$ a smooth function of the metric, by the perturbation theory of simple eigenvalues (4A.3 Calculus and Fixed Points in Banach Spaces).

13.9History

John von Neumann created the theory of unbounded self-adjoint operators in 1929–1932 for quantum mechanics, including the spectral theorem and the theory of deficiency indices; Marshall Stone proved his theorem and developed the theory independently (1929–1932). Kurt Friedrichs introduced his extension in 1934. Franz Rellich (1939) and Tosio Kato (1951) proved the perturbation theorem; Kato's paper on the self-adjointness of atomic Hamiltonians is a landmark of mathematical physics. The min–max principle goes back to Fischer (1905) and Courant (1920), and the Rayleigh–Ritz method to Ritz (1908).

13.10Exercises

Solution

$(v, w) \perp J(u, Au) = (-Au, u)$ iff $-\langle Au, v\rangle + \langle u, w\rangle = 0$ for all $u \in D(A)$, iff $v \in D(A^*)$ with $A^*v = w$. Orthogonal complements are closed. If $\Gamma(A) \subseteq \Gamma(B)$ then $(J\Gamma(B))^\perp \subseteq (J\Gamma(A))^\perp$.

Solution

(a) For $u, v \in H^1$, $\langle-iu', v\rangle - \langle u, -iv'\rangle = -i(u(1)\bar v(1) - u(0)\bar v(0))$. For $u$ in the domain this is $-iu(0)(e^{i\alpha}\bar v(1) - \bar v(0))$, which vanishes for all such $u$ iff $v(1) = e^{i\alpha}v(0)$. So $D(P_\alpha^*) = D(P_\alpha)$ (that $D(P_\alpha^*) \subseteq H^1$ follows as for the closure of the derivative). (b) $-iu' = \lambda u$ gives $u = e^{i\lambda x}$, and the boundary condition gives $e^{i\lambda} = e^{i\alpha}$: $\lambda_k = \alpha + 2\pi k$, $e_k = e^{i(\alpha + 2\pi k)x} = e^{i\alpha x}e^{2\pi ikx}$, an orthonormal basis since the $e^{2\pi ikx}$ are (2B.7 Fourier Series and the First Heat Equation) and multiplication by $e^{i\alpha x}$ is unitary. (c) $\alpha = 0$: periodic boundary conditions, constants.

Solution

$(\operatorname{Ran}G)^\perp = \ker G^* = \ker G = 0$, so the range is dense. The graph of $G^{-1}$ is the graph of $G$ with coordinates swapped, $S\Gamma(G)$ with $S(x, y) = (y, x)$. Then $\Gamma((G^{-1})^*) = (JS\Gamma(G))^\perp$, and $JS(x, y) = (-x, y)$; so $(v, w)$ is in it iff $\langle-x, v\rangle + \langle Gx, w\rangle = 0$ for all $x$, iff $v = G^*w = Gw$, iff $(v, w) \in S\Gamma(G) = \Gamma(G^{-1})$.

Solution

(a) The form norm is the $H^1$ norm, and $C^\infty([0, 1])$ is dense in $H^1(0, 1)$. (b) If $q(u, v) = \langle f, v\rangle$ for all $v \in H^1$, then taking $v \in C_c^\infty$ gives $-u'' + u = f$ weakly, so $u \in H^2$; integrating by parts with general $v$ leaves $u'(1)v(1) - u'(0)v(0) = 0$ for all $v$, so $u'(0) = u'(1) = 0$. The boundary condition is not imposed but natural: it comes out of the variational problem. (c) $-u'' = \lambda u$, $u' = 0$ at both ends: $u = \cos(k\pi x)$, $\lambda = k^2\pi^2$, $k = 0, 1, 2, \dots$. Dirichlet: $\sin(k\pi x)$, $k \geq 1$, same values but starting at $k = 1$. Each Neumann eigenvalue $\lambda_k^N \leq \lambda_k^D$, as min–max predicts ($H^1_0 \subseteq H^1$).

Solution

In $L^2(\mu)$, $U(t)$ is multiplication by $e^{-ita}$, unitary, with $U(t + s) = U(t)U(s)$. For $\phi \in L^2$, $\|U(t)\phi - \phi\|^2 = \int|e^{-ita} - 1|^2|\phi|^2 \to 0$ by dominated convergence. For $a\phi \in L^2$, $\big|i\frac{e^{-ita} - 1}{t} - a\big||\phi| \leq 2|a||\phi|$ (since $|e^{-ix} - 1| \leq |x|$) and tends to $0$ pointwise, so again by dominated convergence.

Solution

$\int_{|\xi|<R}|\hat u| \leq (\frac43\pi R^3)^{1/2}\|u\|_2$. $\int_{|\xi|>R}|\hat u| = \int_{|\xi|>R}|\xi|^{-2}\cdot|\xi|^2|\hat u| \leq \big(\int_{|\xi|>R}|\xi|^{-4}\big)^{1/2}\frac{\|\Delta u\|_2}{4\pi^2} = \big(\frac{4\pi}{R}\big)^{1/2}\frac{\|\Delta u\|_2}{4\pi^2}$. Choose $R$ large so that the second coefficient is below $\epsilon$. The high frequencies are controlled by $\Delta u$ with a constant that becomes small, which is what relative bound $0$ needs.

Solution

(a) $\int|\nabla|w||^2 = \int|\nabla w|^2$ (the gradient of $|w|$ is $\pm\nabla w$ almost everywhere, and $0$ where $w = 0$) and $\int R|w|^2 = \int Rw^2$. (b) $v = |w| \geq 0$ solves $-4\Delta v + (R - \lambda_1)v = 0$, hence $-4\Delta v + (R - \lambda_1 + c)v = cv \geq 0$ with $R - \lambda_1 + c > 0$: a supersolution of an operator with positive zeroth-order term, so if $v$ vanished at a point it would vanish identically (strong maximum principle, 6A.5 Maximum Principles). As $w$ is continuous and $|w| > 0$, $w$ never vanishes, so has constant sign on the connected $M$. (c) If $\lambda_1$ had two independent eigenfunctions, a combination orthogonal to a positive one would be an eigenfunction changing sign, contradicting (b).