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 .
These operators have unique spectral properties, with real-valued spectra and . 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 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- must be carefully chosen to ensure self-adjointness
Maximal domain often determined by solving the equation A∗ψ=Aψ for all ψ in the domain
Symmetric operators with equal 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 ⟨Aϕ,ψ⟩=⟨ϕ,Aψ⟩ for all ϕ,ψ 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
allows representation as an integral over projection-valued measures
Continuous and discrete spectrum possible, with 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 in addition to discrete eigenvalues
Functional calculus
Allows definition of functions of self-adjoint operators using spectral theorem
Continuous 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
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
and deficiency indices play a crucial role in the theory
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=∑i=1nλiPi, where Pi are orthogonal projections
Eigenvalues λi 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 provides a countable basis of eigenvectors
Decomposition takes the form A=∑i=1∞λi⟨⋅,ei⟩ei, where ei 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=∫RλdE(λ), where E(λ) is the spectral measure
Continuous spectrum possible in addition to discrete spectrum
Applications in quantum mechanics for observables like position and momentum operators