5.2 Symmetric operators
7 min read•august 21, 2024
Symmetric operators are the backbone of spectral theory, bridging abstract math and physics. They exhibit special properties in inner product spaces, making them crucial for and mathematical physics applications.
These operators have and , simplifying analysis and interpretation. Understanding symmetric operators, their extensions, and relationships to self-adjoint operators is key for grasping quantum mechanics and advanced spectral theory concepts.
Definition of symmetric operators
- Symmetric operators form a crucial foundation in spectral theory, providing a bridge between abstract linear algebra and physical applications
- These operators exhibit special properties in inner product spaces, making them essential for understanding quantum mechanics and other areas of mathematical physics
Formal vs adjoint operators
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- Formal defined as the operator satisfying for all x and y in the
- Adjoint operator extends the concept of formal adjoint to unbounded operators on Hilbert spaces
- Symmetric operators satisfy for all x and y in the domain, making them a special case of formal adjoint operators
- Distinction between formal and adjoint operators becomes crucial when dealing with unbounded operators
Domain considerations
- Domain of a symmetric operator must be dense in the Hilbert space
- Symmetric operators may have different domains for the operator and its adjoint
- Careful consideration of domains necessary when determining if an operator is truly symmetric
- Extensions of symmetric operators often involve expanding the domain while preserving symmetry
Properties of symmetric operators
- Symmetric operators possess unique characteristics that make them valuable in spectral theory and quantum mechanics
- These properties often simplify calculations and lead to important physical interpretations in various applications
Symmetry in inner product spaces
- Symmetric operators preserve the inner product structure of the space
- For a symmetric operator A, holds for all vectors x and y in the domain
- This property leads to real-valued expectation values in quantum mechanics
- Symmetry ensures that the operator's matrix representation (in orthonormal basis) is Hermitian
Relation to self-adjoint operators
- Every is symmetric, but not every symmetric operator is self-adjoint
- Self-adjoint operators have equal domains for the operator and its adjoint
- Symmetric operators can be extended to become self-adjoint under certain conditions
- applies directly to self-adjoint operators, making them more convenient for analysis
Spectral properties
- Spectral properties of symmetric operators play a crucial role in understanding their behavior and applications
- These properties form the foundation for many important results in spectral theory and quantum mechanics
Real eigenvalues
- All eigenvalues of a symmetric operator are real numbers
- Proof involves using the symmetry property and the definition of eigenvalues
- Real eigenvalues correspond to physically observable quantities in quantum mechanics
- of a symmetric operator may include continuous parts in addition to discrete eigenvalues
Orthogonal eigenvectors
- Eigenvectors corresponding to distinct eigenvalues of a symmetric operator are orthogonal
- This orthogonality property simplifies the analysis of symmetric operators
- Allows for the construction of orthonormal bases using eigenvectors
- In quantum mechanics, orthogonality of eigenvectors relates to the mutual exclusivity of measurement outcomes
Extensions of symmetric operators
- Extensions of symmetric operators are crucial for understanding their complete behavior and applications
- These extensions often bridge the gap between symmetric and self-adjoint operators, which have more robust spectral properties
Friedrichs extension
- provides a method to extend certain symmetric operators to self-adjoint operators
- Involves constructing a new inner product using the original operator
- Guarantees the existence of a self-adjoint extension for semi-
- Widely used in quantum mechanics to define self-adjoint Hamiltonians
von Neumann extension theory
- von Neumann's theory provides a comprehensive framework for understanding all self-adjoint extensions of a symmetric operator
- Utilizes the concept of deficiency subspaces to characterize possible extensions
- Allows for the classification of symmetric operators based on their
- Provides a method to construct all possible self-adjoint extensions of a given symmetric operator
Deficiency indices
- Deficiency indices provide crucial information about the nature of symmetric operators and their possible extensions
- These indices play a central role in von Neumann's extension theory and the classification of symmetric operators
Definition and significance
- Deficiency indices (n+, n-) defined as the dimensions of the kernels of (A* ± iI)
- Measure the "lack of self-adjointness" of a symmetric operator
- Equal deficiency indices (n+ = n-) indicate the possibility of self-adjoint extensions
- Unequal deficiency indices (n+ ≠ n-) imply no self-adjoint extensions exist
Calculation methods
- Solving the equations (A* ± iI)f = 0 to find the deficiency subspaces
- Using boundary conditions for differential operators to determine deficiency indices
- Analyzing the domain and range of the operator to compute the indices
- Employing functional analysis techniques (resolvent equations) for more complex operators
Symmetric vs self-adjoint operators
- Understanding the distinction between symmetric and self-adjoint operators is crucial in spectral theory
- This comparison highlights important differences in domain, spectral properties, and applications
Key differences
- Domain equality (A = A*) required for self-adjoint operators, not necessary for symmetric operators
- Self-adjoint operators always have a complete set of eigenfunctions, symmetric operators may not
- Spectral theorem applies directly to self-adjoint operators, but not always to symmetric operators
- Self-adjoint operators have a unique spectral representation, symmetric operators may lack this property
Conditions for equivalence
- Symmetric operator with equal deficiency indices can be extended to a self-adjoint operator
- Closed symmetric operator with domain equal to that of its adjoint is self-adjoint
- Bounded symmetric operator on a Hilbert space is automatically self-adjoint
- occurs when a symmetric operator has a unique self-adjoint extension
Applications in quantum mechanics
- Symmetric operators play a fundamental role in the mathematical formulation of quantum mechanics
- Their properties align closely with the physical requirements of observable quantities in quantum systems
Observables as symmetric operators
- Physical observables in quantum mechanics represented by symmetric (often self-adjoint) operators
- Symmetry ensures real expectation values, corresponding to measurable quantities
- Eigenstates of symmetric operators represent possible measurement outcomes
- Time evolution in quantum mechanics governed by self-adjoint Hamiltonian operators
Momentum and position operators
- Momentum operator symmetric on suitable domain of functions
- Position operator (multiplication by x) symmetric on L²(ℝ)
- Commutation relation fundamental to quantum mechanics
- Unboundedness of these operators requires careful domain considerations
Cayley transform
- The provides a powerful tool for analyzing symmetric operators
- It establishes a connection between symmetric operators and certain unitary operators, offering new perspectives on operator properties
Definition for symmetric operators
- Cayley transform of a symmetric operator A defined as U = (A - iI)(A + iI)⁻¹
- Inverse Cayley transform given by A = i(I + U)(I - U)⁻¹
- Maps symmetric operators to partial isometries (subset of unitary operators)
- Preserves important spectral properties of the original operator
Properties and applications
- Cayley transform of a self-adjoint operator yields a unitary operator
- Used to study the deficiency indices and self-adjoint extensions of symmetric operators
- Simplifies the analysis of unbounded operators by mapping them to bounded operators
- Provides a method for constructing functional calculus for self-adjoint operators
Bounded symmetric operators
- Bounded symmetric operators form a special class with particularly nice properties
- Their behavior closely resembles that of finite-dimensional symmetric matrices
Spectral theorem
- Spectral theorem guarantees a spectral decomposition for bounded self-adjoint operators
- Allows representation of the operator as an integral with respect to a projection-valued measure
- Enables functional calculus, allowing functions of the operator to be defined
- For compact symmetric operators, the spectral theorem yields a discrete eigenvalue decomposition
Relation to normal operators
- Bounded symmetric operators are a subset of normal operators (AA* = A*A)
- Normal operators have a more general spectral theorem, which reduces to the self-adjoint case for symmetric operators
- Commutator [A, A*] = 0 for symmetric operators, a key property of normal operators
- Study of symmetric operators often extends to normal operators in more advanced spectral theory
Unbounded symmetric operators
- present unique challenges and require more sophisticated analysis
- They are crucial in quantum mechanics and other areas of mathematical physics
Closure and essential self-adjointness
- of a symmetric operator defined as the smallest closed extension
- Essential self-adjointness occurs when the closure of a symmetric operator is self-adjoint
- Criteria for essential self-adjointness involve analyzing the deficiency subspaces
- Important in quantum mechanics for defining observables on physically relevant domains
Defect subspaces
- are the kernels of (A* ± iI) for a symmetric operator A
- Dimension of defect subspaces determines the deficiency indices
- Analysis of defect subspaces crucial for understanding possible self-adjoint extensions
- In quantum mechanics, defect subspaces relate to boundary conditions in certain physical systems
Examples and counterexamples
- Examining specific examples and counterexamples helps solidify understanding of symmetric operators
- These cases illustrate the nuances and potential pitfalls in working with symmetric operators
Finite-dimensional cases
- All symmetric operators in finite dimensions are self-adjoint
- Symmetric matrices represent finite-dimensional symmetric operators
- Diagonalization of symmetric matrices always possible with orthogonal eigenvectors
- Spectral theorem for finite-dimensional symmetric operators yields complete eigenvalue decomposition
Infinite-dimensional cases
- Momentum operator on L²[0,1] symmetric but not self-adjoint without proper boundary conditions
- Laplacian operator on suitable domains provides examples of unbounded self-adjoint operators
- Multiplication operator by x on L²(ℝ) illustrates an unbounded self-adjoint operator
- Shift operator on l²(ℤ) symmetric but not self-adjoint, demonstrating the importance of domain considerations
Key Terms to Review (25)
Adjoint operator: An adjoint operator is a linear operator that corresponds to another operator in a specific way, defined through the inner product in a Hilbert space. The adjoint of an operator captures important properties like symmetry and self-adjointness, making it essential for understanding the structure and behavior of linear operators. The concept of adjoint operators is central to various properties and classifications of operators, influencing their relationships with closed, bounded, and continuous linear operators.
Bounded symmetric operators: Bounded symmetric operators are linear operators on a Hilbert space that are symmetric and also bounded, meaning they have a finite operator norm. These operators play a crucial role in spectral theory, particularly in the study of self-adjoint operators, as they have real eigenvalues and are associated with physical observables in quantum mechanics.
Boundedness: Boundedness refers to a property of operators or functions that limits their output values within a specified range, ensuring that there exists a constant such that the operator or function does not grow indefinitely. This concept is crucial in various contexts, as it implies stability and predictability, particularly when analyzing operators in Hilbert spaces, closed operators, and symmetric operators. Understanding boundedness is key to exploring the resolvent set and determining the continuity and behavior of linear operators.
Cayley Transform: The Cayley transform is a mathematical construct that connects bounded linear operators on a Hilbert space to unitary operators. It is defined for a symmetric operator and is instrumental in spectral theory, particularly in understanding deficiency indices and the properties of symmetric operators.
Closure: Closure refers to the smallest closed set containing a given set in a topological space, or more specifically, the set of all limit points of that set plus the original points. In the context of spectral theory, closure is essential for understanding how operators behave and ensuring that certain properties, like self-adjointness, hold true. Closure helps define the boundaries of operator domains and ensures that symmetric operators can be analyzed effectively.
Compact Operator: A compact operator is a linear operator between Banach spaces that maps bounded sets to relatively compact sets. This means that when you apply a compact operator to a bounded set, the image will not just be bounded, but its closure will also be compact, making it a powerful tool in spectral theory and functional analysis.
Defect Subspaces: Defect subspaces are mathematical structures associated with symmetric operators that measure the extent to which an operator fails to be self-adjoint. They provide a way to understand the limitations of a symmetric operator's domain and its relation to self-adjoint extensions. The dimension of these defect subspaces plays a crucial role in characterizing the operator and its spectral properties.
Deficiency indices: Deficiency indices are integers that characterize the extent to which a symmetric operator fails to be self-adjoint. They provide important information about the solvability of associated differential equations and the existence of self-adjoint extensions. Understanding deficiency indices is crucial when dealing with unbounded operators, as they help determine whether the operator can be extended to a self-adjoint operator and play a key role in spectral theory.
Dirichlet boundary condition: A Dirichlet boundary condition specifies the values a solution must take on the boundary of the domain. This type of condition is crucial for various mathematical and physical problems, allowing one to control the behavior of solutions at the edges of a given region, thus influencing the overall solution of differential equations.
Domain: In mathematics, particularly in functional analysis, the domain refers to the set of all input values (or elements) for which an operator or function is defined. Understanding the domain is crucial as it determines where an operator can act and ensures that the operations performed are valid and meaningful. The concept of domain plays a pivotal role in defining various properties and behaviors of operators, especially when dealing with unbounded self-adjoint operators, symmetric operators, and linear transformations.
Eigenvalue Spectrum: The eigenvalue spectrum refers to the set of eigenvalues associated with a linear operator or matrix. These eigenvalues give insights into the behavior of the operator, such as stability, oscillatory modes, and other important properties in various mathematical contexts. Understanding the eigenvalue spectrum is crucial for analyzing symmetric operators and structures like graph Laplacians, as it reveals significant characteristics of these systems.
Essential self-adjointness: Essential self-adjointness is a property of a symmetric operator in the context of unbounded operators, indicating that the operator has a unique self-adjoint extension that is essential in its domain. When an operator is essentially self-adjoint, it means that its deficiency indices are both zero, which connects closely to the ideas of symmetric operators and adjoint operators. This concept is pivotal in understanding the stability and behavior of quantum mechanical systems where the physical observables are represented by such operators.
Friedrichs extension: Friedrichs extension is a method used to obtain a self-adjoint extension of a symmetric operator defined on a dense domain in a Hilbert space. It plays a crucial role in the spectral theory of operators by ensuring that certain symmetric operators can be extended to self-adjoint operators, which are essential for defining physical observables in quantum mechanics. This extension is particularly significant when considering deficiency indices and essential self-adjointness, as it provides a systematic approach to deal with symmetric operators that might not be initially self-adjoint.
Neumann Boundary Condition: The Neumann boundary condition is a type of boundary condition used in differential equations, particularly in the context of physical problems involving heat conduction or fluid flow. It specifies the value of the derivative of a function on the boundary, indicating how the function behaves at the boundary. This condition is essential in various mathematical frameworks, affecting spectral properties, solutions to differential equations, and the behavior of physical systems.
Orthogonal Eigenvectors: Orthogonal eigenvectors are eigenvectors that correspond to distinct eigenvalues of an operator and are perpendicular to each other in the inner product space. This property is significant because it helps in simplifying many problems in linear algebra and spectral theory, especially when working with symmetric and self-adjoint operators. The orthogonality of these eigenvectors plays a critical role in the diagonalization of matrices and the representation of linear transformations.
Orthogonal Projections: Orthogonal projections refer to the process of mapping a vector onto a subspace such that the resulting vector is the closest point in that subspace to the original vector. This concept is fundamental in linear algebra and plays a critical role in understanding spectral measures, symmetric operators, bounded self-adjoint operators, and the broader context of orthogonality and projections, highlighting how vectors relate within different subspaces.
Quantum Mechanics: Quantum mechanics is a fundamental theory in physics that describes the behavior of matter and energy at very small scales, such as atoms and subatomic particles. This theory introduces concepts such as wave-particle duality, superposition, and entanglement, fundamentally changing our understanding of the physical world and influencing various mathematical and physical frameworks.
Rayleigh's Principle: Rayleigh's Principle is a method used in spectral theory to determine the eigenvalues of a self-adjoint operator by minimizing the Rayleigh quotient over appropriate subspaces. This principle highlights the relationship between the eigenvalues of an operator and variational methods, providing a powerful tool to find approximate solutions and insights into the behavior of symmetric operators.
Real eigenvalues: Real eigenvalues are scalar values associated with a linear operator or matrix that indicate the factor by which an eigenvector is stretched or compressed during the transformation. They are significant because they provide insight into the stability and dynamics of systems described by the operator, linking to essential properties such as symmetry and self-adjointness. Additionally, the presence of real eigenvalues is crucial when analyzing various mathematical structures, such as graphs, where they can illustrate connectivity and other characteristics.
Self-adjoint operator: A self-adjoint operator is a linear operator defined on a Hilbert space that is equal to its own adjoint, meaning that it satisfies the condition $$A = A^*$$. This property ensures that the operator has real eigenvalues and a complete set of eigenfunctions, making it crucial for understanding various spectral properties and the behavior of physical systems in quantum mechanics.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra and functional analysis that characterizes the structure of self-adjoint and normal operators on Hilbert spaces. It establishes that such operators can be represented in terms of their eigenvalues and eigenvectors, providing deep insights into their behavior and properties, particularly in relation to compactness, spectrum, and functional calculus.
Spectrum: In mathematics and physics, the spectrum of an operator is the set of values that describes the behavior of the operator, particularly its eigenvalues. It provides critical insight into the properties and behaviors of systems modeled by operators, revealing how they act on various states or functions.
Unbounded Symmetric Operators: Unbounded symmetric operators are linear operators defined on a dense subset of a Hilbert space that are symmetric but not necessarily bounded. These operators are significant because they can have important implications in quantum mechanics and mathematical physics, often related to the study of self-adjoint extensions. Understanding their properties, such as their domain and the implications of unboundedness, is crucial for grasping the behavior of quantum systems.
Vibrating strings: Vibrating strings refer to the oscillation of a string fixed at both ends, which produces a range of frequencies based on its length, tension, and mass per unit length. The phenomenon of vibrating strings is central to understanding wave mechanics and is a practical example of how symmetric operators function in physics and mathematics.
Von Neumann extension theory: Von Neumann extension theory is a framework in functional analysis that allows the extension of symmetric operators to self-adjoint operators on a Hilbert space. This theory plays a crucial role in understanding the properties of symmetric operators, which may not initially be defined on the entire Hilbert space but can be extended while preserving their essential characteristics. It connects to the spectral properties of operators and provides a structured way to analyze unbounded operators.