🎵Spectral Theory Unit 3 – Spectral Theorem: Bounded Self-Adjoint Operators

Key Concepts and Definitions

• Hilbert spaces vector spaces equipped with an inner product that allows the measurement of lengths and angles
• Bounded linear operators linear maps between Hilbert spaces that have a finite operator norm
• Adjoint operator $A^*$ of a bounded linear operator $A$ satisfies $\langle Ax, y \rangle = \langle x, A^*y \rangle$ for all $x, y$ in the Hilbert space
• Self-adjoint operators bounded linear operators $A$ that are equal to their adjoint $(A = A^*)$
  • Equivalently, $\langle Ax, y \rangle = \langle x, Ay \rangle$ for all $x, y$ in the Hilbert space
• Spectrum of an operator set of all eigenvalues of the operator
• Projection-valued measure (PVM) assigns a projection operator to each measurable subset of the real line
• Spectral measure associated with a self-adjoint operator $A$ is a projection-valued measure $E$ such that $A = \int_{\mathbb{R}} \lambda dE(\lambda)$

Historical Context and Importance

• Developed in the early 20th century by mathematicians such as David Hilbert, John von Neumann, and Hermann Weyl
• Fundamental result in functional analysis and operator theory
• Provides a powerful framework for understanding the structure and properties of self-adjoint operators
• Plays a crucial role in quantum mechanics, where observables are represented by self-adjoint operators
  • Spectral theorem allows for the interpretation of measurement outcomes and the decomposition of states
• Generalizes the concept of diagonalization for matrices to infinite-dimensional Hilbert spaces
• Enables the study of continuous spectra and the resolution of the identity
• Connects the algebraic properties of operators with the geometric properties of Hilbert spaces

Self-Adjoint Operators: Properties and Examples

• Symmetric in the sense that $\langle Ax, y \rangle = \langle x, Ay \rangle$ for all $x, y$ in the Hilbert space
• Have real eigenvalues and orthogonal eigenvectors corresponding to distinct eigenvalues
• Examples in finite dimensions include real symmetric matrices and Hermitian matrices
• Examples in infinite dimensions include:
  • Multiplication operator $(Mf)(x) = xf(x)$ on $L^2(\mathbb{R})$
  • Laplace operator $-\Delta$ on a suitable domain in $L^2(\mathbb{R}^n)$
  • Schrödinger operator $-\Delta + V$ with a real-valued potential $V$
• Continuous functions of self-adjoint operators are also self-adjoint
• Unitary operators $(U^* = U^{-1})$ are closely related to self-adjoint operators through the exponential map $U = e^{iA}$

Spectral Theorem: Statement and Intuition

• States that every self-adjoint operator $A$ on a Hilbert space $\mathcal{H}$ can be represented as an integral with respect to a unique projection-valued measure $E$
  • $A = \int_{\mathbb{R}} \lambda dE(\lambda)$
• Intuitively, the spectral theorem decomposes the operator $A$ into a "continuous sum" of projection operators scaled by real numbers
• The projection-valued measure $E$ assigns a projection operator $E(\Delta)$ to each measurable subset $\Delta$ of the real line
  • $E(\Delta)$ projects onto the subspace of $\mathcal{H}$ corresponding to the part of the spectrum of $A$ contained in $\Delta$
• The spectral measure $E$ satisfies the properties of a projection-valued measure:
  • $E(\emptyset) = 0$ and $E(\mathbb{R}) = I$
  • $E(\Delta_1 \cap \Delta_2) = E(\Delta_1)E(\Delta_2)$ for all measurable sets $\Delta_1, \Delta_2$
  • Countable additivity: $E(\cup_{n=1}^{\infty} \Delta_n) = \sum_{n=1}^{\infty} E(\Delta_n)$ for pairwise disjoint measurable sets $\Delta_n$
• The spectral theorem provides a canonical form for self-adjoint operators and reveals their underlying structure

Proof Outline and Key Steps

1. Show that the resolvent operator $R(\lambda) = (A - \lambda I)^{-1}$ exists for all non-real $\lambda$ and is bounded
2. Prove that the resolvent satisfies the first resolvent identity: $R(\lambda) - R(\mu) = (\mu - \lambda)R(\lambda)R(\mu)$
3. Define the spectral measure $E$ using the Stieltjes inversion formula: $E((a, b]) = \lim_{\varepsilon \to 0^+} \frac{1}{2\pi i} \int_a^b [R(x - i\varepsilon) - R(x + i\varepsilon)] dx$
4. Show that $E$ is a projection-valued measure and that $A = \int_{\mathbb{R}} \lambda dE(\lambda)$
   • Use the properties of the resolvent and the Stieltjes inversion formula
5. Prove the uniqueness of the spectral measure $E$ using the Stone-Weierstrass theorem and the properties of the resolvent
6. Extend the result to unbounded self-adjoint operators using the Cayley transform and the spectral theorem for unitary operators

Applications in Quantum Mechanics

• In quantum mechanics, observables are represented by self-adjoint operators on a Hilbert space
• The spectral theorem allows for the interpretation of measurement outcomes
  • Eigenvalues correspond to possible measurement results
  • Projections onto eigenspaces represent the state of the system after a measurement
• Enables the decomposition of a quantum state into a superposition of eigenstates
• Provides a framework for understanding the time evolution of quantum systems
  • Time evolution operator $U(t) = e^{-iHt/\hbar}$ is unitary, where $H$ is the Hamiltonian (self-adjoint) operator
• Allows for the study of continuous spectra in quantum systems (position, momentum operators)
• Fundamental in the mathematical formulation of quantum mechanics and the development of quantum theory

Computational Methods and Examples

• Numerical computation of eigenvalues and eigenvectors for self-adjoint matrices using algorithms such as the QR algorithm or the Jacobi method
• Finite element methods for approximating the eigenvalues and eigenfunctions of self-adjoint differential operators (Laplace, Schrödinger)
  • Discretize the domain and construct a finite-dimensional subspace
  • Solve the resulting matrix eigenvalue problem
• Example: Computation of energy levels and wavefunctions for a quantum harmonic oscillator
  • Hamiltonian $H = -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + \frac{1}{2}m\omega^2x^2$ is a self-adjoint operator on a suitable domain in $L^2(\mathbb{R})$
  • Discretize using a finite difference scheme or a basis of Hermite functions
  • Solve the resulting matrix eigenvalue problem to obtain approximate energy levels and wavefunctions
• Spectral methods for solving partial differential equations involving self-adjoint operators
  • Expand the solution in terms of eigenfunctions of the operator
  • Leads to efficient and accurate numerical schemes

Related Theorems and Extensions

• Spectral theorem for compact self-adjoint operators
  • Ensures the existence of an orthonormal basis of eigenvectors
  • Eigenvalues can be ordered as a decreasing sequence converging to zero
• Spectral theorem for unitary operators
  • Unitary operators have a spectral decomposition with respect to a projection-valued measure on the unit circle
• Spectral theorem for normal operators $(AA^* = A^*A)$
  • Normal operators have a spectral decomposition with respect to a projection-valued measure on the complex plane
• Functional calculus for self-adjoint operators
  • Allows the definition of functions $f(A)$ of a self-adjoint operator $A$ using the spectral measure
• Spectral theory of unbounded self-adjoint operators
  • Extension of the spectral theorem to unbounded operators using the Cayley transform and the spectral theorem for unitary operators
• Spectral theory of self-adjoint operators on Banach spaces
  • Generalization of the spectral theorem to operators on certain Banach spaces (e.g., $L^p$ spaces) using the notion of a resolution of the identity