7.1 Spectral theorem for unbounded self-adjoint operators

The spectral theorem for unbounded self-adjoint operators extends the concept to operators that aren't necessarily bounded. It provides a unique projection-valued measure that decomposes the Hilbert space into orthogonal subspaces, corresponding to different parts of the operator's spectrum.

This theorem is crucial for quantum mechanics and mathematical physics. It allows for functional calculus of unbounded operators, enabling operations like exponentiation and taking roots. This helps solve operator equations and analyze long-term behavior of quantum systems.

Unbounded Self-adjoint Operators

Definition and Properties

• Unbounded self-adjoint operators act as linear operators defined on dense subspaces of Hilbert spaces without necessarily being bounded
• These operators satisfy specific symmetry conditions while operating on their domains
• Domain of an unbounded self-adjoint operator A denoted as D(A) exists as a proper subspace of the Hilbert space
• Self-adjointness for unbounded operators requires identical domains and actions for both the operator and its adjoint
• Graph of an unbounded self-adjoint operator maintains closure in the product Hilbert space H × H
• Spectrum of unbounded self-adjoint operators can include both discrete and continuous parts unlike bounded operators with only discrete spectra

Applications and Examples

• Unbounded self-adjoint operators play crucial roles in quantum mechanics representing physical observables (position and momentum)
• Momentum operator in quantum mechanics serves as an unbounded self-adjoint operator defined as $P = -iℏ \frac{d}{dx}$ on the domain of absolutely continuous functions
• Position operator in quantum mechanics acts as another unbounded self-adjoint operator defined as $Q = x$ on the domain of square-integrable functions
• Laplacian operator $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ represents an unbounded self-adjoint operator in mathematical physics
• Sturm-Liouville operators, arising in differential equations, form a class of unbounded self-adjoint operators

Spectral Theorem for Unbounded Operators

Statement and Significance

• Spectral theorem for unbounded self-adjoint operators extends the bounded case to encompass unbounded operators
• For an unbounded self-adjoint operator A on a Hilbert space H, a unique projection-valued measure E exists on Borel subsets of R satisfying $A = \int \lambda dE(\lambda)$
• Spectral measure E decomposes the Hilbert space into orthogonal subspaces corresponding to different spectrum parts of A
• Theorem ensures every unbounded self-adjoint operator possesses a spectral representation using projections
• Domain of the unbounded self-adjoint operator characterizes through the spectral measure as $D(A) = \{x \in H : \int \lambda^2 d\langle E(\lambda)x, x\rangle < \infty\}$

Proof and Implications

• Proof of the spectral theorem for unbounded operators typically employs the Cayley transform mapping the unbounded operator to a unitary operator
• Cayley transform defined as $U = (A - iI)(A + iI)^{-1}$ establishes a bijection between unbounded self-adjoint operators and unitary operators with -1 not in the point spectrum
• Spectral theorem allows for the functional calculus of unbounded self-adjoint operators enabling the definition of functions of these operators
• Functional calculus permits operations like exponentiation and taking roots of unbounded self-adjoint operators
• Resolution of the identity associated with an unbounded self-adjoint operator constructs its functional calculus

Applying the Spectral Theorem

Decomposition and Analysis

• Spectral theorem facilitates decomposition of unbounded self-adjoint operators into direct integrals of multiplication operators
• For any Borel function f, the operator f(A) defines as $f(A) = \int f(\lambda) dE(\lambda)$ where E represents the spectral measure of A
• Spectral decomposition enables analysis of the operator's spectrum including point spectrum (eigenvalues), continuous spectrum, and residual spectrum
• Spectral projections obtained from the spectral measure decompose the Hilbert space into invariant subspaces of the operator
• Invariant subspaces correspond to different parts of the spectrum (discrete eigenvalues or continuous spectrum intervals)

Applications and Problem Solving

• Spectral theorem applications include solving operator equations, analyzing long-term behavior of quantum systems, and studying differential equation stability
• Operator equations like $Ax = y$ solve using the spectral decomposition as $x = \int \frac{1}{\lambda} dE(\lambda)y$ when 0 does not belong to the spectrum of A
• Long-term behavior of quantum systems analyzes through the time evolution operator $U(t) = e^{-itA}$ where A represents the Hamiltonian
• Stability of differential equations studies by examining the spectrum of the associated unbounded operator
• Heat equation $\frac{\partial u}{\partial t} = \Delta u$ solves using the spectral decomposition of the Laplacian operator