5.6 Spectral theory of unbounded self-adjoint operators

Spectral theory of unbounded self-adjoint operators extends the analysis of bounded operators to infinite-dimensional spaces. It's crucial for modeling physical systems with unbounded observables in quantum mechanics, requiring careful consideration of domain and self-adjointness properties.

The spectral theorem, a cornerstone of this theory, provides a decomposition of self-adjoint operators into spectral components. It's a fundamental tool for analyzing unbounded operators in quantum mechanics, generalizing the diagonalization of matrices to infinite-dimensional spaces.

Unbounded self-adjoint operators

• Fundamental concept in spectral theory extends analysis of bounded operators to infinite-dimensional spaces
• Crucial for modeling physical systems with unbounded observables in quantum mechanics
• Requires careful consideration of domain and self-adjointness properties

Definition and properties

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• Unbounded linear operator $A$ on Hilbert space $H$ with dense domain $D(A)$
• Self-adjointness condition $\langle Ax, y \rangle = \langle x, Ay \rangle$ for all $x, y \in D(A)$
• Spectrum $\sigma(A)$ consists of real values due to self-adjointness
• May have continuous spectrum in addition to discrete eigenvalues

Domain and range

• Domain $D(A)$ dense subset of Hilbert space $H$
• Range $R(A)$ may not be all of $H$, unlike bounded operators
• Closedness of operator important for spectral analysis
• Graph of operator $G(A) = \{(x, Ax) : x \in D(A)\}$ closed in $H \times H$

Symmetric vs self-adjoint operators

• Symmetric operator satisfies $\langle Ax, y \rangle = \langle x, Ay \rangle$ for $x, y \in D(A)$
• Self-adjoint operator additionally requires $D(A) = D(A^*)$
• Distinction crucial for spectral properties and physical applications
• Self-adjoint extensions of symmetric operators (von Neumann theory)

Spectral theorem

• Cornerstone of spectral theory generalizes diagonalization of matrices
• Provides decomposition of self-adjoint operators into spectral components
• Fundamental tool for analyzing unbounded operators in quantum mechanics

Statement for unbounded operators

• For self-adjoint operator $A$, spectral measure $E$ exists on Borel sets of $\mathbb{R}$
• Operator $A$ represented as $A = \int_{\mathbb{R}} \lambda dE(\lambda)$
• Spectral measure $E$ projects onto eigenspaces for discrete spectrum
• For continuous spectrum, $E$ gives generalized eigenfunctions

Spectral decomposition

• Hilbert space $H$ decomposed into direct sum of spectral subspaces
• $H = H_p \oplus H_c \oplus H_{sc}$ (pure point, continuous, singular continuous)
• Each vector $\psi \in H$ uniquely expressed as $\psi = \psi_p + \psi_c + \psi_{sc}$
• Spectral measure $E$ determines projection onto each subspace

Continuous vs discrete spectrum

• Discrete spectrum consists of isolated eigenvalues (bound states)
• Continuous spectrum corresponds to scattering states in physics
• Mixed spectra possible, combining discrete and continuous parts
• Spectral type determines long-time behavior of quantum systems

Resolvent and spectrum

• Resolvent operator central tool for analyzing spectral properties
• Connects operator theory to complex analysis
• Spectrum classified based on behavior of resolvent

Resolvent operator

• For $z \notin \sigma(A)$, resolvent defined as $R(z) = (A - zI)^{-1}$
• Analytic operator-valued function on resolvent set $\rho(A)$
• Spectral mapping theorem relates spectrum of $A$ to that of $R(z)$
• Resolvent identity: $R(z) - R(w) = (w-z)R(z)R(w)$ for $z,w \in \rho(A)$

Point spectrum vs continuous spectrum

• Point spectrum $\sigma_p(A)$ consists of eigenvalues
• Continuous spectrum $\sigma_c(A)$ where $R(z)$ exists but unbounded
• Residual spectrum $\sigma_r(A)$ (rare for self-adjoint operators)
• Spectrum $\sigma(A) = \sigma_p(A) \cup \sigma_c(A) \cup \sigma_r(A)$

Essential spectrum

• Part of spectrum stable under compact perturbations
• Includes continuous spectrum and accumulation points of eigenvalues
• Characterizes behavior of operator "at infinity"
• Weyl's essential spectrum theorem for self-adjoint operators

Functional calculus

• Extends functions of real variables to functions of operators
• Powerful tool for manipulating and analyzing self-adjoint operators
• Connects spectral theory to measure theory and integration

Borel functional calculus

• For Borel function $f$ and self-adjoint $A$, define $f(A) = \int_{\mathbb{R}} f(\lambda) dE(\lambda)$
• Extends polynomial functional calculus to wider class of functions
• Preserves algebraic properties (linearity, multiplication)
• Allows rigorous definition of functions like $e^{iA}$ (crucial for quantum dynamics)

Spectral projections

• For Borel set $\Omega \subset \mathbb{R}$, spectral projection $E(\Omega) = \chi_\Omega(A)$
• Projects onto subspace corresponding to spectrum in $\Omega$
• Orthogonal projections satisfying $E(\Omega_1)E(\Omega_2) = E(\Omega_1 \cap \Omega_2)$
• Generate spectral measure: $E(\Omega) = \int_\Omega dE(\lambda)$

Spectral measure

• Projection-valued measure on Borel sets of $\mathbb{R}$
• For $\psi \in H$, scalar measure $\mu_\psi(\Omega) = \langle \psi, E(\Omega)\psi \rangle$
• Spectral theorem expressed as $\langle \psi, A\psi \rangle = \int_{\mathbb{R}} \lambda d\mu_\psi(\lambda)$
• Determines operator uniquely up to unitary equivalence

Applications in quantum mechanics

• Spectral theory provides mathematical foundation for quantum mechanics
• Self-adjoint operators represent physical observables
• Spectrum corresponds to possible measurement outcomes

Observables as self-adjoint operators

• Position $Q$, momentum $P$, and Hamiltonian $H$ represented by unbounded self-adjoint operators
• Canonical commutation relations: $[Q, P] = i\hbar I$ (on suitable domain)
• Measurement postulate: spectrum of observable gives possible measurement outcomes
• Expectation values given by $\langle A \rangle = \langle \psi, A\psi \rangle$ for normalized state $\psi$

Energy spectra

• Spectrum of Hamiltonian $H$ represents possible energy values
• Discrete spectrum corresponds to bound states (atoms, quantum wells)
• Continuous spectrum represents scattering states (free particles)
• Negative eigenvalues indicate bound states, positive continuous spectrum for scattering

Position and momentum operators

• Position operator $Q$ multiplication by $x$ in position representation
• Momentum operator $P = -i\hbar \frac{d}{dx}$ in position representation
• Both have continuous spectrum $\mathbb{R}$ (unbounded observables)
• Fourier transform relates position and momentum representations

Perturbation theory

• Studies how spectrum changes under small modifications to operator
• Essential for analyzing realistic physical systems
• Combines spectral theory with analytic techniques

Kato-Rellich theorem

• Provides conditions for self-adjointness of perturbed operator
• For self-adjoint $A$ and symmetric $B$, if $B$ relatively bounded with respect to $A$
• Then $A + B$ self-adjoint on $D(A)$ if relative bound $< 1$
• Ensures spectral theory applies to perturbed operator

Analytic perturbation theory

• Studies dependence of eigenvalues and eigenvectors on perturbation parameter
• Power series expansions: $\lambda(\varepsilon) = \lambda_0 + \varepsilon \lambda_1 + \varepsilon^2 \lambda_2 + \cdots$
• Resolvent expansion techniques for continuous spectrum
• Applications to atomic physics (Stark effect, Zeeman effect)

Stability of essential spectrum

• Essential spectrum typically stable under compact perturbations
• Weyl's theorem: $\sigma_{ess}(A + K) = \sigma_{ess}(A)$ for compact $K$
• Important for scattering theory and solid-state physics
• Allows focus on discrete spectrum changes in many applications

Deficiency indices

• Characterize extent to which symmetric operator fails to be self-adjoint
• Key concept in theory of self-adjoint extensions
• Important for boundary value problems and quantum mechanics

Symmetric extensions

• For symmetric operator $A$, consider $A^* \supset A$
• Deficiency subspaces $K_\pm = \ker(A^* \mp iI)$
• Deficiency indices $(n_+, n_-)$ dimensions of $K_+$ and $K_-$
• Symmetric extensions exist if $n_+ + n_- > 0$

von Neumann's theorem

• Characterizes all self-adjoint extensions of symmetric operator
• Self-adjoint extensions exist iff $n_+ = n_-$
• Extensions parameterized by unitary maps $U : K_+ \to K_-$
• Each extension corresponds to different boundary condition

Self-adjoint extensions

• Physical importance in quantum mechanics (different boundary conditions)
• Examples: radial Schrödinger operator on $[0, \infty)$ with different behaviors at origin
• Krein-von Neumann extension (soft boundary condition)
• Friedrichs extension (hard boundary condition)

Spectral analysis techniques

• Methods for determining spectral properties of concrete operators
• Combine functional analysis, complex analysis, and operator theory
• Essential for applications in mathematical physics

Weyl's criterion

• Characterizes essential spectrum of self-adjoint operator
• $\lambda \in \sigma_{ess}(A)$ iff exists sequence $\{\psi_n\}$ with $\|\psi_n\| = 1$, $\psi_n \rightharpoonup 0$
• And $\|(A - \lambda I)\psi_n\| \to 0$ as $n \to \infty$
• Powerful tool for analyzing Schrödinger operators

RAGE theorem

• Relates spectral types to long-time behavior of quantum systems
• For $\psi \in H_{ac}$, time average of $\langle \psi(t), B\psi(t) \rangle$ vanishes for compact $B$
• Continuous spectrum associated with "escaping" behavior
• Applications in scattering theory and quantum chaos

Mourre theory

• Provides criteria for absolute continuity of spectrum
• Based on commutator methods and positive commutator estimates
• $i[H, A] \geq \theta E_\Delta(H)$ for some $\theta > 0$ and spectral projection $E_\Delta(H)$
• Applications to multi-particle quantum systems and dispersive estimates

Absolutely continuous spectrum

• Part of continuous spectrum associated with "scattering states"
• Characterized by existence of absolutely continuous spectral measure
• Important for understanding long-time behavior of quantum systems

Scattering theory connection

• Absolutely continuous subspace $H_{ac}$ associated with scattering states
• Wave operators $W_\pm = s-\lim_{t \to \pm\infty} e^{itH}e^{-itH_0}P_{ac}(H_0)$
• Intertwining property: $HW_\pm = W_\pm H_0$
• Scattering operator $S = W_+^* W_-$ relates incoming and outgoing states

Wave operators

• Relate dynamics of perturbed system to free system
• Existence and completeness of wave operators key questions
• Kato-Rosenblum theorem for trace-class perturbations
• Stationary methods based on resolvent comparisons

Absolutely continuous subspace

• $H_{ac}$ spanned by vectors with absolutely continuous spectral measure
• Orthogonal to pure point and singular continuous subspaces
• Characterizes "scattering states" in quantum mechanics
• Cyclic decomposition theorem for absolutely continuous subspace

Singular continuous spectrum

• "Exotic" part of spectrum, neither pure point nor absolutely continuous
• Often associated with fractal structures and unusual quantum dynamics
• Challenging to analyze, requires sophisticated mathematical tools

Cantor-type spectra

• Prototypical example of singular continuous spectrum
• Appears in quasiperiodic Schrödinger operators (almost Mathieu operator)
• Cantor set structure in spectrum reflects quasiperiodic potential
• Spectral properties depend sensitively on parameters (Hofstadter's butterfly)

Quantum dynamics

• Intermediate behavior between localization and transport
• Power-law decay of time-averaged probabilities
• Fractal dimensions of spectral measure relate to dynamical properties
• Guarneri-Combes-Last theorem connects spectral and dynamical fractal dimensions

Anderson localization

• Transition between extended and localized states in disordered systems
• Pure point spectrum with exponentially decaying eigenfunctions in strongly disordered regime
• Absolutely continuous spectrum for weak disorder in high dimensions
• Possibility of singular continuous spectrum at mobility edge or in one-dimensional systems