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Self-adjoint operator

In functional analysis, a self-adjoint operator on a complex Hilbert space H is a densely defined linear operator T: \mathcal{D}(T) \to H, where \mathcal{D}(T) is a dense subspace of H, such that T = T^* and \mathcal{D}(T) = \mathcal{D}(T^*), with T^* denoting the adjoint operator defined by \langle Tx, y \rangle = \langle x, T^* y \rangle for all x \in \mathcal{D}(T) and y \in \mathcal{D}(T^*).^[1] This condition implies that T is symmetric, meaning \langle Tx, y \rangle = \langle x, Ty \rangle for all x, y \in \mathcal{D}(T), and closed.^[2] Self-adjoint operators generalize Hermitian matrices from finite-dimensional spaces to infinite-dimensional Hilbert spaces and can be bounded (with \mathcal{D}(T) = H) or unbounded (with proper dense domains).^[3] For bounded self-adjoint operators, the spectrum lies on the real line, eigenvalues are real, and eigenspaces for distinct eigenvalues are orthogonal.^[2] The spectral theorem for self-adjoint operators provides a functional calculus, allowing them to be diagonalized in an appropriate sense via a resolution of the identity, which is essential for analyzing their behavior.^[1] In quantum mechanics, self-adjoint operators represent physical observables such as position, momentum, and energy, ensuring that measurement outcomes (eigenvalues) are real numbers and that expectation values \langle \psi | T | \psi \rangle are real for normalized states \psi.^[4] Unbounded self-adjoint operators, like the Hamiltonian -\Delta + V on L^2(\mathbb{R}^n), model differential operators and require careful domain specifications to ensure self-adjointness.^[1] A symmetric operator is essentially self-adjoint if it has a unique self-adjoint extension, which is crucial for uniqueness in physical models.^[1]

Definitions and Basic Concepts

Formal Definition

A self-adjoint operator on a Hilbert space is a fundamental concept in functional analysis, generalizing the notion of Hermitian matrices to infinite-dimensional settings. Consider a complex Hilbert space H equipped with an inner product \langle \cdot, \cdot \rangle. A linear operator A: D(A) \to H, where D(A) \subset H is a dense linear subspace (the domain of A), is said to be densely defined. The graph of A is the set G(A) = \{ (x, Ax) \mid x \in D(A) \} \subset H \oplus H, and A is closed if G(A) is a closed subspace of H \oplus H.^[5] The adjoint operator A^* of a densely defined linear operator A is defined as follows: the domain D(A^*) = \{ y \in H \mid \exists z \in H \text{ such that } \langle Ax, y \rangle = \langle x, z \rangle \ \forall x \in D(A) \}, and A^* y = z for y \in D(A^*). An operator A is symmetric (or Hermitian) if D(A) \subset D(A^*) and \langle Ax, y \rangle = \langle x, A^* y \rangle = \langle x, Ay \rangle for all x, y \in D(A). A densely defined operator A is self-adjoint if it is symmetric and D(A) = D(A^*), which implies that A is closed.^[5]^[6] The concept of self-adjoint operators originated in David Hilbert's foundational work on integral equations around 1906–1910, where he developed the abstract framework of Hilbert spaces and symmetric operators to address spectral problems in infinite dimensions.^[7]

Symmetric Operators

In the context of a Hilbert space H, a densely defined linear operator A: D(A) \subset H \to H is called symmetric if it satisfies the condition \langle Ax, y \rangle = \langle x, Ay \rangle for all x, y \in D(A).^[8] This inner product equality ensures that the operator preserves the sesquilinear form on its domain. The symmetry condition implies a specific relation to the adjoint operator A^*: A \subseteq A^*, meaning D(A) \subseteq D(A^*) and Ax = A^*x for all x \in D(A).^[8] In general, the inclusion D(A) \subseteq D(A^*) may be proper, so the domains differ, distinguishing symmetric operators from self-adjoint ones where equality holds. Equivalently, symmetry can be characterized via graphs: the graph G(A) = \{ (x, Ax) \mid x \in D(A) \} \subseteq H \times H satisfies G(A) \subseteq G(A^*).^[9] A concrete example is the momentum operator P = -i \frac{d}{dx} defined on the domain D(P) = C_c^\infty(\mathbb{R}) (smooth compactly supported functions) in the space L^2(\mathbb{R}); this operator is symmetric but not self-adjoint, as D(P) is a proper subset of D(P^*).^[10] Symmetric operators play a foundational role as precursors to self-adjoint operators, with every self-adjoint operator being symmetric, though the converse fails in general.^[8]

Properties of Bounded Self-Adjoint Operators

Key Properties

A bounded self-adjoint operator A on a Hilbert space is normal, meaning it commutes with its adjoint: AA^* = A^*A. Since A = A^* by definition, this simplifies to A^2 = A^2, confirming the normality condition.^[11] The eigenvalues of a bounded self-adjoint operator are always real. Specifically, if Ax = \lambda x for some eigenvector x \neq 0, then \lambda \in \mathbb{R}, as the self-adjoint property ensures that \langle Ax, x \rangle = \overline{\langle x, Ax \rangle}, implying \lambda = \overline{\lambda}.^[12] Eigenspaces corresponding to distinct eigenvalues of a bounded self-adjoint operator are orthogonal. For eigenvalues \lambda \neq \mu, if Ax = \lambda x and Ay = \mu y, then \langle x, y \rangle = 0, following from the self-adjoint relation \langle Ax, y \rangle = \langle x, Ay \rangle, which yields (\lambda - \mu) \langle x, y \rangle = 0.^[12] A symmetric operator on a Hilbert space that is bounded (and thus defined on the entire space) is self-adjoint, by the Hellinger–Toeplitz theorem, which guarantees that such an operator extends continuously and satisfies A = A^*.^[6] For a bounded self-adjoint operator A, the operator norm equals the spectral radius: \|A\| = \sup \{ |\lambda| : \lambda \in \sigma(A) \}, where \sigma(A) is the spectrum of A. This equality holds because self-adjoint operators are normal, and for normal operators, the spectral radius formula aligns with the norm.^[6] Bounded symmetric operators are self-adjoint provided they are closed; since bounded operators on Hilbert spaces have closed graphs when defined everywhere, this criterion reinforces that symmetric boundedness implies self-adjointness.^[6]

Norm and Resolvent Estimates

For a bounded linear operator A on a Hilbert space \mathcal{H}, the resolvent set \rho(A) is defined as the set of all complex numbers \lambda \in \mathbb{C} such that A - \lambda I is bijective and its inverse (A - \lambda I)^{-1} is bounded.^[13] For a bounded self-adjoint operator A, the resolvent set satisfies \rho(A) = \mathbb{C} \setminus \sigma(A), where \sigma(A) denotes the spectrum of A, and moreover \sigma(A) \subseteq \mathbb{R}.^[14]^[15] A key quantitative property of the resolvent for self-adjoint operators arises from their normality. Specifically, for \lambda \notin \sigma(A),

\|(A - \lambda I)^{-1}\| = \frac{1}{\operatorname{dist}(\lambda, \sigma(A))},

where \operatorname{dist}(\lambda, \sigma(A)) = \inf_{\mu \in \sigma(A)} |\lambda - \mu|.^[16] This equality provides a precise bound that is instrumental in perturbation theory and stability analysis, as it directly ties the growth of the resolvent norm to the distance from the real spectrum. The numerical range of a bounded self-adjoint operator A, defined as

W(A) = \left\{ \langle Ax, x \rangle : x \in \mathcal{H}, \|x\| = 1 \right\},

coincides exactly with the convex hull of its spectrum, \operatorname{conv}(\sigma(A)).^[17] Since \sigma(A) \subseteq \mathbb{R}, W(A) is a closed interval [\inf \sigma(A), \sup \sigma(A)], which encapsulates the possible values of quadratic forms and aids in estimating operator behavior without full spectral knowledge. Self-adjointness also imposes structure on quadratic forms. For any x \in \mathcal{H}, the inner product \langle Ax, x \rangle is real-valued, reflecting the reality of the spectrum.^[14] Furthermore, by the Cauchy-Schwarz inequality applied to the sesquilinear form,

|\langle Ax, x \rangle| \leq \|A\| \|x\|^2,

with equality achievable when A attains its spectral radius.^[18] In fact, the operator norm itself is given by

\|A\| = \sup_{\|x\|=1} |\langle Ax, x \rangle|,

offering a variational characterization useful for numerical approximations and bounds in applications.^[18]

Spectrum of Self-Adjoint Operators

Real Spectrum

The spectrum of a self-adjoint operator A on a Hilbert space H, denoted \sigma(A), is the set of all complex numbers \lambda \in \mathbb{C} such that A - \lambda I does not have a bounded inverse in the algebra of bounded linear operators on H.^[14] A fundamental property of self-adjoint operators is that their spectrum lies entirely on the real line, i.e., \sigma(A) \subseteq \mathbb{R}. To see this, suppose \lambda = s + it with t \neq 0. For any nonzero u \in \mathcal{D}(A), \operatorname{Im} \langle (A - \lambda I)u, u \rangle = -t \|u\|^2 \neq 0, so if (A - \lambda I)u = 0 then contradiction, implying A - \lambda I is injective. Moreover, |\operatorname{Im} \langle (A - \lambda I)u, u \rangle| \leq \|(A - \lambda I)u\| \|u\| implies \|(A - \lambda I)u\| \geq |t| \|u\| for u \in \mathcal{D}(A), so the range is closed. The range is also dense, since its orthogonal complement is \ker(A - \overline{\lambda} I) = \{0\} by a similar argument for \overline{\lambda}. Thus, A - \lambda I is bijective with bounded inverse (of norm at most $1/|t|). Hence, no non-real \lambda belongs to \sigma(A).^[14]^[19]^[20] The spectrum of any operator on a complex Banach space partitions into the point spectrum \sigma_p(A) (eigenvalues), the continuous spectrum \sigma_c(A), and the residual spectrum \sigma_r(A). For self-adjoint operators, the residual spectrum is empty: \sigma_r(A) = \emptyset. If \lambda \in \sigma_r(A), then A - \lambda I is injective but its range is not dense, so there exists nonzero y \in H orthogonal to the range, leading to \overline{\lambda} being an eigenvalue of A. Since A is self-adjoint, this implies \lambda is an eigenvalue, contradicting injectivity. Thus, \sigma(A) = \sigma_p(A) \cup \sigma_c(A).^[21]^[19] The spectrum of a self-adjoint operator coincides with its approximate point spectrum: \lambda \in \sigma(A) if and only if \inf \{ \|(A - \lambda I)x\| / \|x\| : x \in \mathcal{D}(A), x \neq 0 \} = 0. Equivalently, there exists a sequence of unit vectors \{x_n\} \subset \mathcal{D}(A) such that \|(A - \lambda I)x_n\| \to 0. This characterization holds because self-adjoint operators have no residual spectrum, and the approximate point spectrum captures both eigenvalues and points in the continuous spectrum. Moreover, any approximate eigenvalue must be real, as \langle A x_n, x_n \rangle is real for unit vectors, implying \lambda is real in the limit.^[22] For bounded self-adjoint operators, the spectrum is a nonempty compact subset of \mathbb{R}. Boundedness ensures \sigma(A) \subseteq [-\|A\|, \|A\|], a closed and bounded interval, hence compact; nonemptiness follows from the spectral radius formula or the fact that \|A\| \in \sigma(A).^[14]^[19]

Continuous and Residual Spectrum

For a self-adjoint operator A on a Hilbert space H, the spectrum \sigma(A) decomposes into the disjoint union of the point spectrum \sigma_p(A), the continuous spectrum \sigma_c(A), and the residual spectrum \sigma_r(A). The point spectrum \sigma_p(A) consists of all \lambda \in \mathbb{C} such that \ker(A - \lambda I) \neq \{0\}, i.e., the eigenvalues of A with corresponding eigenvectors in \mathcal{D}(A). These eigenvalues are real, as \sigma(A) \subseteq \mathbb{R} for self-adjoint A.^[23] The continuous spectrum \sigma_c(A) comprises those \lambda \in \sigma(A) for which A - \lambda I is injective (so \lambda \notin \sigma_p(A)), the range \ran(A - \lambda I) is dense in H, but \ran(A - \lambda I) \neq H. In this case, \lambda acts as an approximate eigenvalue: there exists a sequence of unit vectors \{u_n\} \subset \mathcal{D}(A) with \|(A - \lambda I)u_n\| \to 0, but no actual eigenvector. For bounded self-adjoint operators, this implies the resolvent (A - \lambda I)^{-1} (where defined) is unbounded. The residual spectrum \sigma_r(A), defined as the set of \lambda \in \sigma(A) where A - \lambda I is injective but \ran(A - \lambda I) is not dense in H, is empty for self-adjoint operators. This emptiness arises because self-adjointness ensures that for real \lambda, the orthogonal complement of \ran(A - \lambda I) coincides with \ker(A - \overline{\lambda} I) = \ker(A - \lambda I); thus, injectivity implies density of the range.^[23]^[24] A canonical example of a purely continuous spectrum occurs with the bounded self-adjoint multiplication operator M_x on L^2[0,1] defined by (M_x f)(t) = t f(t) for f \in L^2[0,1]. Here, \sigma(M_x) = [0,1], \sigma_c(M_x) = [0,1], and \sigma_p(M_x) = \emptyset, since for any \lambda \in [0,1], M_x - \lambda I is injective (no non-zero f satisfies (\lambda - t)f(t) = 0 almost everywhere on a set of positive measure), the range is dense (approximable by step functions), but not surjective (e.g., the function $1/(\lambda - t + i\epsilon) cannot be exactly hit for small \epsilon > 0).^[19] For unbounded self-adjoint operators, the continuous spectrum admits a characterization via Weyl's criterion: \lambda \in \sigma(A) if and only if there exists an orthonormal sequence \{u_n\} \subset \mathcal{D}(A) (a Weyl sequence) such that \|(A - \lambda I)u_n\| \to 0. Specifically, \lambda \in \sigma_c(A) when such a sequence exists with no weak limit in \ker(A - \lambda I) (ensuring injectivity) and the sequence weakly tends to zero, distinguishing it from the point spectrum. This criterion is particularly useful for operators like differential operators, where the continuous spectrum often fills intervals on the real line corresponding to scattering states.^[24]^[23]

Spectral Theorem

General Statement

The spectral theorem for self-adjoint operators provides a canonical decomposition of such operators on a Hilbert space, representing them as integrals with respect to projection-valued measures. Specifically, for a self-adjoint operator A on a separable Hilbert space H, there exists a unique (up to equivalence) projection-valued measure E: \mathcal{B}(\mathbb{R}) \to \mathcal{B}(H), where \mathcal{B}(\mathbb{R}) denotes the Borel \sigma-algebra on \mathbb{R} and \mathcal{B}(H) the bounded linear operators on H, such that

A = \int_{\mathbb{R}} \lambda \, dE(\lambda),

with the integral interpreted in the strong operator topology.^[25] This resolution E satisfies the key commutativity property A E(\Delta) = E(\Delta) A for every Borel set \Delta \subseteq \mathbb{R}, ensuring that the projections E(\Delta) commute with A and thus preserve its self-adjoint structure.^[25] The uniqueness of the spectral measure E holds up to equivalence of projection-valued measures, meaning that if another measure E' satisfies the same integral representation for A, then E(\Delta) = E'(\Delta) for all Borel sets \Delta outside a set of measure zero with respect to both.^[25] For bounded self-adjoint operators A, the support of E is contained within the interval [-\|A\|, \|A\|], reflecting the fact that the spectrum of A lies in this bounded interval.^[25] This abstract formulation of the spectral theorem was established by John von Neumann in 1932, who proved it using the double commutant theorem for bounded operators on Hilbert space, extending earlier work on normal operators and providing a rigorous foundation for spectral decompositions in quantum mechanics.

Multiplication Operator Form

The multiplication operator form of the spectral theorem provides a concrete realization of self-adjoint operators through unitary equivalence to multiplication operators on suitable L^2 spaces. For a self-adjoint operator A on a separable Hilbert space H, there exists a measure space (\Sigma, \mu) and a unitary operator U: H \to \int^\oplus_{\Sigma} H(\lambda) \, d\mu(\lambda), where each H(\lambda) is a Hilbert space, such that U A U^{-1} acts as multiplication by the identity function \lambda on the direct integral space.^[26] This equivalence implies that A is unitarily equivalent to \int^\oplus_{\Sigma} M_\lambda \, d\mu(\lambda), where M_\lambda denotes the multiplication operator by \lambda on H(\lambda).^[26] In the case of bounded self-adjoint operators, the direct integral can be represented as a direct sum \bigoplus_{n=1}^\infty L^2(\mathbb{R}, d\mu_n) over at most countable multiplicity, where only finitely many summands are non-trivial if the multiplicity is finite.^[26] This structure simplifies the representation while preserving the spectral properties of A. The isomorphism arises via a change of variables induced by the spectral projections, which map elements of H to functions on the spectrum through the projection-valued measure E(\lambda) associated with A.^[26] A proof sketch relies on the decomposition of H into cyclic subspaces. For each cyclic subspace generated by a vector under powers of A, a unitary map is constructed to L^2 of the spectral measure, extending to the full space by direct sum over an orthogonal basis of cyclic vectors; the self-adjointness ensures the multiplication function is real-valued.^[26] This form yields the integral representation \langle A x, y \rangle = \int \lambda \, d\langle E(\lambda) x, y \rangle for x, y \in H, where E(\lambda) is the spectral family of projections.^[26] The projection measure E(\lambda) from the general spectral theorem statement underpins this construction, resolving A into its spectral components.^[26]

Functional Calculus

The Borel functional calculus provides a framework for associating Borel measurable functions f: \mathbb{R} \to \mathbb{C} with a self-adjoint operator A on a Hilbert space, defining a new operator f(A) through the spectral resolution of A. Specifically, if E is the spectral measure associated to A via the spectral theorem, then f(A) = \int_{\sigma(A)} f(\lambda) \, dE(\lambda), where \sigma(A) is the spectrum of A. This operator is densely defined on the domain \{ \xi \in H \mid \int_{\sigma(A)} |f(\lambda)|^2 \, d\|E(\lambda) \xi\|^2 < \infty \} and extends continuously if f is bounded.^[27] Key properties of this calculus include the preservation of self-adjointness: if f is real-valued, then f(A) is self-adjoint. Additionally, the operator norm satisfies \|f(A)\| \leq \sup_{\lambda \in \sigma(A)} |f(\lambda)|, with equality holding when f is continuous and bounded on \sigma(A). The calculus extends the polynomial functional calculus, where for a polynomial p(\lambda) = \sum_k a_k \lambda^k, p(A) = \sum_k a_k A^k is defined via the power series, and \sigma(p(A)) = p(\sigma(A)).^[27] One important application is the representation of the resolvent operator: for z \notin \sigma(A), the resolvent (A - zI)^{-1} = \int_{\sigma(A)} (\lambda - z)^{-1} \, dE(\lambda). In quantum mechanics, the functional calculus is essential for constructing the time evolution operator e^{-itA}, where A is the Hamiltonian, enabling the solution of the time-dependent Schrödinger equation via the unitary group generated by A.^[27]

Symmetric vs Self-Adjoint Operators

Core Differences

A symmetric operator A on a Hilbert space is defined as a densely defined linear operator satisfying A \subset A^*, where A^* is the adjoint operator, meaning that for all x, y \in D(A), \langle Ax, y \rangle = \langle x, A^* y \rangle holds, with D(A) \subseteq D(A^*).^[28] In contrast, a self-adjoint operator requires A = A^*, which implies not only the symmetry condition but also that the domains coincide, D(A) = D(A^*).^[29] This equality of domains is the core distinction, as symmetric operators may have proper extensions to self-adjoint ones, whereas self-adjoint operators are maximal symmetric.^[28] Symmetric operators are closable, meaning their closure \overline{A} is also symmetric, and the graph of a closed symmetric operator is closed in the product space H \oplus H.^[29] However, not all closed symmetric operators are self-adjoint; self-adjointness demands the stricter condition D(A) = D(A^*), ensuring no further symmetric extensions exist.^[28] For instance, in infinite-dimensional spaces, many differential operators are symmetric but require domain enlargement via boundary conditions to achieve self-adjointness.^[29] The existence of self-adjoint extensions for a closed symmetric operator A is determined by the deficiency subspaces K_+ = \ker(A^* - iI) and K_- = \ker(A^* + iI), with deficiency indices n_+ = \dim K_+ and n_- = \dim K_-.^[28] Such extensions exist if and only if n_+ = n_-, and the operator is self-adjoint precisely when both indices are zero.^[29] These indices quantify the "deficiency" in the domain that prevents symmetry from implying self-adjointness.^[28] For positive symmetric operators (those with \langle Ax, x \rangle \geq 0 for all x \in D(A)), self-adjoint extensions always exist, with the Friedrichs extension A_F and Krein (or Krein-von Neumann) extension A_K serving as extremal cases.^[30] The Friedrichs extension is the "maximal" one in the sense of quadratic forms, often arising from completing the form domain, while the Krein extension is the "minimal" positive extension, and all other positive self-adjoint extensions A' satisfy A_K \leq A' \leq A_F in the form sense.^[30] These extensions, developed by Friedrichs and Krein, highlight how domain choices resolve the gap between symmetric and self-adjoint operators for applications like quantum mechanics.^[30]

Essential Self-Adjointness

A symmetric operator A on a Hilbert space is said to be essentially self-adjoint if its closure \overline{A} is self-adjoint.^[31] This property ensures that \overline{A} provides a unique self-adjoint extension of A.^[32] The essential self-adjointness of a symmetric operator A is determined by its deficiency indices, defined as n_\pm = \dim \ker(A^* \mp iI), where A^* is the adjoint of A.^[31] Specifically, A is essentially self-adjoint if and only if both deficiency indices vanish, i.e., n_+ = n_- = 0.^[32] A fundamental theorem states that if A is a symmetric operator and \ker(A^* \pm iI) = \{0\}, then A is essentially self-adjoint, with its closure \overline{A} coinciding with A^*.^[31] In this case, the unique self-adjoint extension is given by \overline{A}.^[32] When the deficiency indices are equal but positive, self-adjoint extensions exist but are not unique, parametrized by unitary operators between the deficiency subspaces.^[31] This contrasts with the essential self-adjoint case, where the zero indices guarantee uniqueness. A classic example is the operator -\frac{d^2}{dx^2} defined on the domain C_c^\infty(\mathbb{R}) in L^2(\mathbb{R}), which is symmetric and essentially self-adjoint.^[32]

Extensions and Deficiency Indices

For a closed symmetric operator A defined on a Hilbert space \mathcal{H}, the deficiency indices are defined as n_\pm = \dim \ker(A^* \mp iI), where A^* is the adjoint of A and I denotes the identity operator.^[33] These indices characterize the possible self-adjoint extensions of A: self-adjoint extensions exist if and only if n_+ = n_-.^[33] Von Neumann's theorem provides a complete parametrization of these extensions. Specifically, the self-adjoint extensions of A are in one-to-one correspondence with the unitary operators U: \ker(A^* - iI) \to \ker(A^* + iI). For each such unitary U, the corresponding self-adjoint extension A_U has domain

D(A_U) = \{ x + y + U y \mid x \in D(A),\ y \in \ker(A^* - iI) \}

and acts as

A_U(x + y + U y) = A x + i y - i U y.

^[34] This construction ensures that A_U is self-adjoint and extends A, with the graph of A_U obtained by "gluing" the graph of A to the deficiency subspaces via U.^[33] When the deficiency indices satisfy n_+ = n_- = 0, the deficiency subspaces are trivial, implying that A is essentially self-adjoint: its closure \overline{A} is self-adjoint, providing a unique self-adjoint extension.^[33] In the case where n_+ = n_- = n < \infty with n > 0, the unitaries form the unitary group U(n), yielding infinitely many distinct self-adjoint extensions.^[33] For n = 1, the extensions are parametrized by a phase \theta \in [0, 2\pi), where U y = e^{i \theta} y for y in a basis of the one-dimensional \ker(A^* - iI), and the domain takes the form above with this U.^[33]

Examples

Finite-Dimensional Operators

In finite-dimensional Hilbert spaces over the complex numbers, specifically \mathbb{C}^n equipped with the standard inner product, a bounded linear operator A is self-adjoint if and only if it is represented by a Hermitian matrix, meaning A = A^*, where A^* is the adjoint operator, or equivalently, the matrix entries satisfy a_{ji} = \overline{a_{ij}} for all i, j = 1, \dots, n.^[3]^[35] A fundamental property of such operators is their diagonalizability: every Hermitian matrix A can be unitarily diagonalized, so there exists a unitary matrix U (satisfying U^* U = I) and a real diagonal matrix D = \operatorname{diag}(\lambda_1, \dots, \lambda_n) such that

A = U D U^*,

where the \lambda_k are the eigenvalues of A.^[35]^[36] This decomposition follows from the spectral theorem for Hermitian matrices, ensuring that the eigenvalues are real and that the corresponding eigenvectors form an orthonormal basis for \mathbb{C}^n.^[37] The spectrum of a self-adjoint operator in this setting consists solely of its real eigenvalues, with no continuous or residual spectrum, and the operator admits an orthonormal basis of eigenvectors. This real spectrum and orthogonal eigenspaces underpin applications in quantum mechanics, where observables are modeled by such operators.^[35]^[37] Functional calculus for a self-adjoint operator A is defined via its spectral decomposition: for a function f: \mathbb{R} \to \mathbb{C} continuous on the spectrum of A, the operator f(A) is given by

f(A) = U f(D) U^*,

where f(D) = \operatorname{diag}(f(\lambda_1), \dots, f(\lambda_n)), ensuring f(A) is also self-adjoint when f is real-valued.^[36]^[37] A prominent example arises in quantum mechanics with the Pauli matrices, which represent the spin operators for a spin-$1/2 particle and are inherently self-adjoint. These 2×2 Hermitian matrices are

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

each with eigenvalues \pm 1 and orthonormal eigenvectors forming a basis for \mathbb{C}^2.^[38]^[39]

Differential Operators on Intervals

A prominent example of a self-adjoint differential operator on a finite interval is the negative second derivative operator -\frac{d^2}{dx^2} acting on L^2[0, \pi], realized with Dirichlet boundary conditions u(0) = u(\pi) = 0. This realization is defined on the domain H^2[0, \pi] \cap H_0^1[0, \pi], where H^2 and H_0^1 denote the standard Sobolev spaces, ensuring the operator is bounded below and self-adjoint.^[40]^[41] The minimal operator associated with -\frac{d^2}{dx^2} is the closure of the densely defined operator on smooth functions with compact support in (0, \pi), which is symmetric but not self-adjoint. Its adjoint, known as the maximal operator, has a larger domain consisting of all functions in H^2[0, \pi] without boundary restrictions, allowing for integration by parts to reveal boundary terms that confirm the deficiency indices are (2, 2). Self-adjoint extensions of this minimal operator are obtained by imposing suitable boundary conditions that make the boundary form vanish, such as Dirichlet conditions u(0) = u(\pi) = 0, Neumann conditions u'(0) = u'(\pi) = 0, periodic conditions u(0) = u(\pi) and u'(0) = u'(\pi), or mixed conditions like u(0) = 0 and u'(\pi) = 0. These extensions parameterize a U(2) family, ensuring the operator is self-adjoint on L^2[0, \pi].^[40]^[42] For the Dirichlet realization on [0, L], the spectrum is purely discrete and consists of eigenvalues \lambda_n = \left(\frac{n\pi}{L}\right)^2 for n = 1, 2, \dots, with corresponding eigenfunctions \sin\left(\frac{n\pi x}{L}\right). These eigenvalues arise from solving the eigenvalue equation -\frac{d^2 u}{dx^2} = \lambda u subject to the boundary conditions, yielding a complete orthonormal basis for L^2[0, L]. On [0, \pi], this simplifies to \lambda_n = n^2.^[43]^[41] In contrast, the first-order momentum operator -i \frac{d}{dx} on L^2[0, 1], with the minimal domain of smooth compactly supported functions in (0, 1), is symmetric but not self-adjoint, as its adjoint has domain H^1[0, 1] and deficiency indices (1, 1). Self-adjoint extensions require boundary conditions linking the values at the endpoints, such as periodic conditions u(0) = u(1) or more generally twisted conditions u(1) = e^{i\theta} u(0) for \theta \in [0, 2\pi), which parameterize the family of self-adjoint realizations. Without such conditions, the operator lacks self-adjointness on the finite interval.^[44]

Schrödinger Operators with Singular Potentials

Schrödinger operators of the form -\Delta + V, where V is a singular potential, present significant challenges to self-adjointness due to the potential's lack of regularity near singularities, which can lead to multiple possible self-adjoint extensions or failure of essential self-adjointness on smooth compactly supported domains.^[45] In quantum mechanics, ensuring self-adjointness is crucial because it guarantees real eigenvalues corresponding to observable energies and unitary time evolution, preserving probability conservation.^[46] A prominent example is the hydrogen atom Hamiltonian H = -\Delta - \frac{1}{|x|} in three dimensions, defined initially on C_c^\infty(\mathbb{R}^3). This operator is essentially self-adjoint, meaning its closure is self-adjoint and unique, despite the Coulomb potential's singularity at the origin; this result follows from Hardy's inequality and perturbation theory, ensuring a well-defined spectrum without needing additional boundary conditions.^[45] In contrast, more singular potentials like V(x) = -\frac{1}{x^2} in one dimension lead to non-essential self-adjointness. For the operator -\frac{d^2}{dx^2} - \frac{\alpha}{x^2} on (0, \infty) with domain C_c^\infty((0,\infty)), the deficiency indices are n_\pm = [1](/page/1) when \alpha \geq \frac{[3](/page/3)}{4}, indicating that the operator is symmetric but not essentially self-adjoint, and requires a one-parameter family of self-adjoint extensions specified by a boundary condition at x=0 to achieve self-adjointness.^[47] For \alpha = [1](/page/1), corresponding to V = -\frac{1}{x^2}, this pathology arises because solutions to the equation (-\frac{d^2}{dx^2} - \frac{1}{x^2} \pm i)\psi = 0 exhibit oscillatory behavior near zero that is square-integrable, leading to non-zero deficiency subspaces.^[47] Kato's theorem provides criteria for essential self-adjointness in higher dimensions or less singular cases: if V is locally in L^2(\mathbb{R}^n) and |V| is relatively bounded with respect to -\Delta by a constant less than 1 (i.e., \| |V| f \| \leq \epsilon \| (-\Delta + 1) f \| + C \| f \| for \epsilon < 1), then -\Delta + V is essentially self-adjoint on C_c^\infty(\mathbb{R}^n).^[45] This applies to potentials like the Coulomb term but fails for highly singular ones like -\frac{1}{x^2} in low dimensions, where relative boundedness breaks down.^[45]

Advanced Topics in Spectral Theory

Spectral Multiplicity

In the spectral decomposition of a self-adjoint operator A on a separable Hilbert space, the multiplicity function m(\lambda) at a point \lambda in the spectrum \sigma(A) is defined as the dimension of the fiber Hilbert space \mathcal{H}(\lambda) in the direct integral representation \int^\oplus_{\sigma(A)} \mathcal{H}(\lambda) \, d\mu(\lambda), where \mu is the spectral measure.^[26] This function m: \sigma(A) \to \mathbb{N} \cup \{\infty\} is measurable and captures the "degeneracy" or dimension of the eigenspaces generalized to continuous spectrum, with m(\lambda) finite or infinite depending on the operator's structure.^[24] For points \lambda not in the spectrum, m(\lambda) = 0.^[26] A special case arises when the multiplicity is discrete, meaning m(\lambda) = k for some fixed finite integer k > 0 almost everywhere with respect to the spectral measure \mu.^[26] In this scenario, the Hilbert space decomposes as a direct sum of k copies of a single L^2(\sigma(A), \mu)-type space, and the operator A is unitarily equivalent to multiplication by \lambda on \bigoplus_{n=1}^k L^2(\mathbb{R}, d\mu).^[26] Discrete multiplicity simplifies the analysis, as the operator's action reduces to k identical copies of a multiplication operator, often seen in finite-rank or finite-dimensional settings.^[24] The spectral measure \mu of A admits a Lebesgue decomposition \mu = \mu_{pp} + \mu_{ac} + \mu_{sc}, where \mu_{pp} is the pure point part supported on the discrete eigenvalues, \mu_{ac} is absolutely continuous with respect to Lebesgue measure (corresponding to band structure in physical models), and \mu_{sc} is singular continuous (arising in fractal or aperiodic systems).^[24] Each component inherits a multiplicity function: for the pure point part, m_{pp}(\lambda) is the geometric multiplicity \dim \ker(A - \lambda I) at eigenvalues \lambda; for the absolutely continuous part, m_{ac}(\lambda) reflects the dimension of the subspace where the spectral measure has a density; and for the singular continuous part, m_{sc}(\lambda) describes the multiplicity in the singular support, often infinite in non-separable cases but finite or countably infinite in separable Hilbert spaces.^[26] The Hilbert space decomposes orthogonally as H = H_{pp} \oplus H_{ac} \oplus H_{sc}, with A restricting to each invariant subspace according to the respective multiplicities.^[24] The spectral type of a self-adjoint operator is fully determined by its multiplicity function m(\lambda) and the equivalence class of the spectral measure \mu under mutual absolute continuity, classifying operators up to unitary equivalence.^[26] Operators with the same spectral type share identical spectral projections and functional calculus properties, enabling comparison of their dynamical or scattering behaviors.^[24] This classification extends the discrete eigenvalue picture to continuous spectra, where the type encodes whether the spectrum is of pure point, absolutely continuous, singular continuous, or mixed nature, modulated by the varying or constant multiplicity.^[26] For compact self-adjoint operators on infinite-dimensional separable Hilbert spaces, the spectrum is purely discrete (except possibly at zero), consisting of a countable set of eigenvalues \{\lambda_n\} accumulating only at zero, each with finite uniform multiplicity m(\lambda_n) = k_n < \infty.^[24] In this case, the spectral measure is a sum of Dirac deltas \mu = \sum_n m(\lambda_n) \delta_{\lambda_n}, and the operator admits an orthonormal basis of eigenvectors, with the finite multiplicity at each \lambda_n determining the dimension of the corresponding eigenspace.^[26]

Direct Integral Decomposition

The direct integral provides a fundamental framework for decomposing Hilbert spaces and self-adjoint operators in spectral theory, allowing for the realization of multiplicity in a measurable manner. A direct integral Hilbert space is constructed over a standard measure space (X, \mu) with a measurable field of Hilbert spaces \{K(x)\}_{x \in X}, where the space H = \int^\oplus_{X, \mu} K(x) \, d\mu(x) consists of measurable sections \psi: X \to \bigcup K(x) such that \psi(x) \in K(x) almost everywhere and \int_X \|\psi(x)\|_{K(x)}^2 \, d\mu(x) < \infty, equipped with the inner product \langle \psi, \phi \rangle_H = \int_X \langle \psi(x), \phi(x) \rangle_{K(x)} \, d\mu(x).^[48] This construction ensures H is a separable Hilbert space when the field admits a countable measurable orthonormal basis.^[49] Operators on such a direct integral space are classified as decomposable or multiplication operators. A decomposable operator T on H acts pointwise almost everywhere via a measurable field of operators \{T(x)\}_{x \in X}, satisfying (T \psi)(x) = T(x) \psi(x) for almost every x \in X, where each T(x) is a bounded operator on K(x) forming a measurable field in the strong sense.^[48] In contrast, multiplication operators act by scalar multiplication on the sections, typically by a measurable function m: X \to \mathbb{C}, so (M_m \psi)(x) = m(x) \psi(x). Decomposable operators converge in the strong operator topology and form an algebra closed under certain operations, facilitating the analysis of commutants in von Neumann algebras.^[49] The spectral theorem for a self-adjoint operator A on a separable Hilbert space H is realized through a unitary equivalence to a multiplication operator on a direct integral space. Specifically, there exists a measure space (\mathbb{R}, \nu) and a measurable field of Hilbert spaces \{H(\lambda)\}_{\lambda \in \mathbb{R}} such that H is unitarily equivalent to \int^\oplus_{\mathbb{R}, \nu} H(\lambda) \, d\nu(\lambda), and A corresponds to multiplication by the identity function \lambda on this space, i.e., (A \psi)(\lambda) = \lambda \psi(\lambda) almost everywhere with respect to \nu.^[50] This decomposition captures the spectral measure of A, where the support of \nu relates to the spectrum \sigma(A), often taken as L^2(\mathbb{R}, H(\lambda), d\mu) for a suitable measure \mu.^[48] More generally, the disintegration theorem extends this to arbitrary representations by allowing a direct integral decomposition over unitary operators. Any separable Hilbert space H admits a disintegration into a direct integral \int^\oplus_X K(x) \, d\mu(x) via a measurable family of unitaries intertwining the structure, enabling the decomposition of operators affiliated to a von Neumann algebra into direct integrals of simpler components.^[49] This framework unifies discrete, continuous, and mixed spectra under a single measurable construction. The multiplicity in this decomposition is captured by the measurable multiplicity function m(\lambda), which describes the dimension of the fibers H(\lambda) and can vary across the spectrum; for instance, in the singular continuous spectrum, m(\lambda) may be infinite on sets of positive measure while remaining finite elsewhere, reflecting the intricate structure of the operator's spectral type.^[50] This multiplicity function provides a precise measure of degeneracy, linking back to the broader concept of spectral multiplicity in self-adjoint operator theory.

Structure of the Laplacian

The negative Laplacian operator -\Delta, defined initially on the Schwartz space \mathcal{S}(\mathbb{R}^d) and extended by closure to a self-adjoint operator on L^2(\mathbb{R}^d), has an absolutely continuous spectrum equal to [0, \infty) with infinite multiplicity.^[51] This spectrum arises because the operator lacks discrete eigenvalues, as confirmed by the existence of Weyl sequences for every \lambda \in [0, \infty), establishing the essential spectrum as the entire non-negative real line.^[52] The Fourier transform provides a unitary equivalence between L^2(\mathbb{R}^d) and itself, under which the negative Laplacian -\Delta is transformed into the multiplication operator by |\xi|^2 (up to a constant factor depending on the Fourier convention, such as $4\pi^2 |\omega|^2).^[51] In this representation, the generalized eigenfunctions are the plane waves e^{i \xi \cdot x}, which satisfy (-\Delta) e^{i \xi \cdot x} = |\xi|^2 e^{i \xi \cdot x} but do not belong to L^2(\mathbb{R}^d), underscoring the continuous nature of the spectrum.^[52] The spectral decomposition of a function f \in L^2(\mathbb{R}^d) is then given by the Fourier inversion formula f(x) = \int_{\mathbb{R}^d} \hat{f}(\xi) e^{i \xi \cdot x} \, d\xi, where \hat{f} is the Fourier transform of f.^[51] To reveal the multiplicity structure explicitly, consider the Fourier domain in spherical coordinates, where \xi = r \theta with r \in \mathbb{R}_+ and \theta \in S^{d-1}. Under this change of variables, the Lebesgue measure d\xi becomes r^{d-1} \, dr \, d\sigma(\theta), where d\sigma is the surface measure on the unit sphere S^{d-1}. Thus, L^2(\mathbb{R}^d, d\xi) is unitarily equivalent to L^2(S^{d-1}, d\sigma) \otimes L^2(\mathbb{R}_+, r^{d-1} \, dr), and the multiplication by |\xi|^2 = r^2 acts as the identity on the first factor and multiplication by r^2 on the second.^[51] This decomposition highlights the infinite multiplicity: for each energy level \lambda > 0, the corresponding "eigenspace" is infinite-dimensional, parametrized by the (d-1)-dimensional sphere S^{d-1} at radius \sqrt{\lambda}.^[52] In contrast, when the negative Laplacian is considered on a bounded domain with appropriate boundary conditions, the spectrum becomes purely discrete, consisting of a sequence of eigenvalues accumulating only at infinity, each with finite multiplicity determined by the domain's geometry.^[51]

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