2.6 Self-adjoint operators

Self-adjoint operators are crucial in spectral theory, extending Hermitian matrices to infinite-dimensional spaces. They're key in quantum mechanics and functional analysis, providing a mathematical framework for physical observables.

These operators have unique spectral properties, with real-valued spectra and orthogonal eigenvectors. The spectral theorem allows for their complete characterization, generalizing matrix diagonalization to infinite-dimensional spaces.

Definition of self-adjoint operators

• Self-adjoint operators form a crucial subset of linear operators in spectral theory, generalizing the concept of Hermitian matrices to infinite-dimensional spaces
• These operators play a fundamental role in quantum mechanics and functional analysis, providing a mathematical framework for describing physical observables

Formal vs bounded operators

• Formal operators defined on a dense subspace of a Hilbert space without considering boundedness
• Bounded operators have a finite operator norm, ensuring continuity across the entire Hilbert space
• Unbounded self-adjoint operators require careful domain considerations to maintain self-adjointness
• Examples include differential operators (unbounded) and multiplication operators (can be bounded or unbounded)

Domain considerations

• Domain of a self-adjoint operator must be carefully chosen to ensure self-adjointness
• Maximal domain often determined by solving the equation $A^* \psi = A \psi$ for all $\psi$ in the domain
• Symmetric operators with equal deficiency indices can be extended to self-adjoint operators
• Domain must be dense in the Hilbert space and closed under the action of the operator

Symmetry vs self-adjointness

• Symmetric operators satisfy $\langle A\phi, \psi \rangle = \langle \phi, A\psi \rangle$ for all $\phi, \psi$ in the domain
• Self-adjoint operators require equality of domains: $Dom(A) = Dom(A^*)$
• Self-adjointness implies symmetry, but the converse does not always hold
• Counterexamples include momentum operator on a finite interval without proper boundary conditions

Properties of self-adjoint operators

• Self-adjoint operators exhibit unique spectral properties that make them invaluable in quantum mechanics and functional analysis
• These properties allow for a comprehensive understanding of the operator's behavior and its associated physical observables

Spectral properties

• Spectrum of a self-adjoint operator consists entirely of real numbers
• Spectral decomposition theorem allows representation as an integral over projection-valued measures
• Continuous and discrete spectrum possible, with point spectrum corresponding to eigenvalues
• Resolvent set contains all complex numbers with non-zero imaginary part

Eigenvalue characteristics

• All eigenvalues of a self-adjoint operator are real
• Eigenvectors corresponding to distinct eigenvalues are orthogonal
• Finite-dimensional self-adjoint operators always have a complete set of eigenvectors
• Infinite-dimensional case may have continuous spectrum in addition to discrete eigenvalues

Functional calculus

• Allows definition of functions of self-adjoint operators using spectral theorem
• Continuous functional calculus extends to bounded Borel functions
• Enables computation of exponentials, square roots, and other functions of operators
• Applications in quantum mechanics for defining observables and time evolution operators

Self-adjoint extensions

• Self-adjoint extensions provide a way to complete symmetric operators to full self-adjoint operators
• This process is crucial in quantum mechanics for defining physically meaningful observables

Symmetric operator extensions

• Symmetric operators with equal deficiency indices can be extended to self-adjoint operators
• Extensions not unique in general, leading to different physical interpretations
• Cayley transform used to characterize all possible self-adjoint extensions
• Examples include momentum operator on an interval with various boundary conditions

von Neumann theory

• Characterizes all self-adjoint extensions of a given symmetric operator
• Deficiency subspaces and deficiency indices play a crucial role in the theory
• Unitary operators between deficiency subspaces correspond to self-adjoint extensions
• Applications in quantum mechanics for defining observables with different boundary conditions

Friedrichs extension

• Provides a canonical self-adjoint extension for semibounded symmetric operators
• Constructed using the associated quadratic form of the symmetric operator
• Guarantees the smallest form domain among all self-adjoint extensions
• Often used in quantum mechanics for defining Hamiltonians with potential terms

Spectral theorem for self-adjoint operators

• Spectral theorem forms the cornerstone of spectral theory, providing a complete characterization of self-adjoint operators
• This theorem generalizes the diagonalization of Hermitian matrices to infinite-dimensional spaces

Finite-dimensional case

• Every self-adjoint operator on a finite-dimensional Hilbert space has a complete set of orthonormal eigenvectors
• Spectral decomposition takes the form $A = \sum_{i=1}^n \lambda_i P_i$, where $P_i$ are orthogonal projections
• Eigenvalues $\lambda_i$ are real and correspond to the spectrum of the operator
• Applications in linear algebra and quantum mechanics for finite-dimensional systems

Compact operators

• Compact self-adjoint operators have a discrete spectrum with 0 as the only possible accumulation point
• Spectral theorem for compact operators provides a countable basis of eigenvectors
• Decomposition takes the form $A = \sum_{i=1}^\infty \lambda_i \langle \cdot, e_i \rangle e_i$, where $e_i$ are eigenvectors
• Examples include integral operators with continuous kernels on bounded domains

Unbounded operators

• Spectral theorem for unbounded self-adjoint operators involves projection-valued measures
• Decomposition takes the form $A = \int_\mathbb{R} \lambda dE(\lambda)$, where $E(\lambda)$ is the spectral measure
• Continuous spectrum possible in addition to discrete spectrum
• Applications in quantum mechanics for observables like position and momentum operators