Glossary

5.4 Essential self-adjointness

7 min read•august 21, 2024

is a key concept in spectral theory, ensuring unique self-adjoint extensions of symmetric operators. It's crucial for studying unbounded operators in quantum mechanics and mathematical physics, providing a foundation for analyzing complex systems.

This topic explores criteria for essential self-adjointness, including and von Neumann's theorem. It also covers examples of essentially self-adjoint operators, methods for proving this property, and its applications in quantum mechanics and spectral analysis.

Definition of essential self-adjointness

  • Essential self-adjointness plays a crucial role in spectral theory by ensuring unique self-adjoint extensions of symmetric operators
  • Provides a foundation for studying unbounded operators in quantum mechanics and other areas of mathematical physics

Symmetric vs self-adjoint operators

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  • Symmetric operators satisfy ⟨Ax,y⟩=⟨x,Ay⟩\langle Ax, y \rangle = \langle x, Ay \rangle for all x, y in the domain of A
  • Self-adjoint operators require equality of domains: D(A)=D(A∗)D(A) = D(A^*)
  • Symmetric operators may have multiple self-adjoint extensions
  • Self-adjoint operators have real spectra and spectral decompositions

Closure of symmetric operators

  • of a A denoted by Aˉ\bar{A}
  • Defined as the smallest closed extension of A
  • Closure preserves symmetry but not necessarily self-adjointness
  • Relationship between closure and adjoint: Aˉ⊂A∗\bar{A} \subset A^*

Domain of self-adjointness

  • Essential self-adjointness occurs when the closure of A equals its adjoint: Aˉ=A∗\bar{A} = A^*
  • Domain of self-adjointness is the largest subspace where A is self-adjoint
  • For essentially self-adjoint operators, this domain is dense in the
  • Uniqueness of self-adjoint extension guaranteed for essentially self-adjoint operators

Criteria for essential self-adjointness

  • Essential self-adjointness provides a powerful tool for analyzing unbounded operators in spectral theory
  • Criteria help determine when a symmetric operator has a unique self-adjoint extension

Deficiency indices

  • Deficiency indices (n+, n-) measure the dimension of deficiency subspaces
  • Defined as n±=dim⁡(ker⁡(A∗∓iI))n_± = \dim(\ker(A^* ∓ iI))
  • Operator A is essentially self-adjoint if and only if n+ = n- = 0
  • Equal deficiency indices (n+ = n-) indicate existence of self-adjoint extensions

von Neumann's theorem

  • Characterizes self-adjoint extensions of symmetric operators
  • States that all self-adjoint extensions of A are in one-to-one correspondence with unitary maps from ker⁡(A∗−iI)\ker(A^* - iI) to ker⁡(A∗+iI)\ker(A^* + iI)
  • Provides a method for constructing self-adjoint extensions
  • Essential self-adjointness occurs when both deficiency subspaces are trivial

Weyl's limit point-limit circle criterion

  • Applies to second-order differential operators on half-line [0, ∞)
  • Limit point case corresponds to essential self-adjointness
  • Limit circle case indicates need for boundary conditions at infinity
  • Criterion based on behavior of solutions to −y′′+qy=λy-y'' + qy = λy as x → ∞

Examples of essentially self-adjoint operators

  • Essential self-adjointness appears frequently in quantum mechanics and spectral theory
  • Understanding these examples helps in applying the concept to more complex systems

Momentum operator

  • Defined as P=−iℏddxP = -i\hbar \frac{d}{dx} on suitable domain in L²(ℝ)
  • Essentially self-adjoint on the space of smooth functions with compact support
  • Unique self-adjoint extension is the closure of P
  • Spectrum consists of entire real line

Position operator

  • Multiplication operator (Qψ)(x) = xψ(x) on L²(ℝ)
  • Essentially self-adjoint on the domain of smooth functions with compact support
  • Unique self-adjoint extension is the maximal multiplication operator
  • Spectrum is also the entire real line

Harmonic oscillator Hamiltonian

  • Defined as H=−ℏ22md2dx2+12mω2x2H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + \frac{1}{2}mω^2x^2 on L²(ℝ)
  • Essentially self-adjoint on the domain of smooth functions with compact support
  • Unique self-adjoint extension has discrete spectrum En=ℏω(n+12)E_n = \hbar ω(n + \frac{1}{2})
  • Eigenfunctions are Hermite polynomials multiplied by Gaussian factors

Methods for proving essential self-adjointness

  • Various techniques exist to establish essential self-adjointness of operators
  • These methods are crucial in spectral theory for understanding operator properties

Nelson's analytic vector theorem

  • Provides sufficient condition for essential self-adjointness
  • Analytic vectors are defined by convergence of power series ∑n=0∞∣∣Anψ∣∣n!tn\sum_{n=0}^∞ \frac{||A^n ψ||}{n!} t^n
  • If A has a dense set of analytic vectors, it is essentially self-adjoint
  • Powerful tool for proving essential self-adjointness of many physical operators

Kato-Rellich theorem

  • Applies to perturbations of self-adjoint operators
  • States that if A is self-adjoint and B is symmetric with relative bound < 1, then A + B is self-adjoint
  • Useful for proving self-adjointness of with potentials
  • Extends to essential self-adjointness under certain conditions

Commutator methods

  • Utilize commutator relations between operators to prove essential self-adjointness
  • Often involve finding a suitable positive operator C such that [A,[A,C]][A, [A, C]] is bounded relative to C
  • Virial theorem and its generalizations play a key role
  • Particularly useful for many-body quantum systems

Applications in quantum mechanics

  • Essential self-adjointness ensures well-defined physical observables in quantum theory
  • Provides mathematical foundation for studying quantum systems on unbounded domains

Unbounded observables

  • Many important physical observables (position, momentum, energy) are unbounded operators
  • Essential self-adjointness guarantees unique self-adjoint extensions
  • Allows for consistent definition of expectation values and measurement outcomes
  • Crucial for maintaining probabilistic interpretation of quantum mechanics

Hamiltonians on infinite domains

  • Essential self-adjointness of Hamiltonians ensures unique time evolution
  • Important for studying scattering problems and systems with continuous spectra
  • Examples include free particle Hamiltonian and Coulomb potential
  • Weyl's limit point criterion often used to establish essential self-adjointness

Schrödinger operators

  • General form H=−Δ+V(x)H = -Δ + V(x) on L²(ℝⁿ)
  • Essential self-adjointness depends on properties of potential V(x)
  • Kato class potentials preserve essential self-adjointness of -Δ
  • Singular potentials may require careful analysis (Coulomb potential)

Relationship to spectral theory

  • Essential self-adjointness forms a crucial link between operator theory and spectral analysis
  • Provides foundation for studying spectra of unbounded operators in physics and mathematics

Spectral theorem for self-adjoint operators

  • Generalizes diagonalization of Hermitian matrices to infinite-dimensional spaces
  • States that every has a spectral decomposition
  • For essentially self-adjoint operators, spectral theorem applies to unique self-adjoint extension
  • Allows representation of operator as integral with respect to projection-valued measure

Functional calculus

  • Enables definition of functions of self-adjoint operators
  • Based on spectral theorem and Borel
  • For essentially self-adjoint A, f(A) well-defined for suitable functions f
  • Important for defining exponential of operators (unitary groups)

Resolvent set vs spectrum

  • Resolvent set consists of complex λ for which (A - λI)⁻¹ exists as
  • Spectrum is complement of resolvent set
  • For essentially self-adjoint operators, spectrum of closure is real
  • Resolvent set crucial for studying spectral properties and perturbation theory

Extensions of symmetric operators

  • When an operator is not essentially self-adjoint, various extensions may exist
  • Understanding these extensions is crucial for applications in physics and mathematics

Friedrichs extension

  • Constructed using quadratic form associated with symmetric operator
  • Always exists for semibounded symmetric operators
  • Represents "hard" extension (largest domain)
  • Preserves lower bound of original operator

Krein-von Neumann extension

  • Represents "soft" extension (smallest domain)
  • Exists for semibounded symmetric operators
  • Minimizes domain while preserving symmetry
  • May introduce additional spectrum at bottom of original spectrum

Comparison of extensions

  • Friedrichs and Krein-von Neumann extensions form extremal self-adjoint extensions
  • All other self-adjoint extensions lie "between" these two
  • Choice of extension can affect physical properties of system
  • In essentially self-adjoint case, all extensions coincide

Boundary conditions and self-adjointness

  • Boundary conditions play crucial role in determining self-adjointness of differential operators
  • Essential self-adjointness often related to absence of need for boundary conditions

Sturm-Liouville problems

  • Second-order differential equations of form (py′)′+qy=λwy(py')' + qy = λwy on interval [a,b]
  • Self-adjointness depends on behavior of solutions near endpoints
  • Regular problems require boundary conditions at both endpoints
  • Singular problems may be essentially self-adjoint without additional conditions

Dirichlet vs Neumann conditions

  • Dirichlet conditions specify function values at boundaries (y(a) = y(b) = 0)
  • Neumann conditions specify derivative values at boundaries (y'(a) = y'(b) = 0)
  • Both lead to self-adjoint realizations of Sturm-Liouville operators
  • Choice of conditions affects spectrum and eigenfunctions

Robin boundary conditions

  • Generalize Dirichlet and Neumann conditions
  • Take form αy(a) + βy'(a) = 0 (similar at b)
  • Lead to self-adjoint realizations for appropriate choices of α and β
  • Allow modeling of more complex physical situations (partial reflection)

Essential self-adjointness in physics

  • Concept of essential self-adjointness has profound implications for physical theories
  • Ensures mathematical consistency of quantum mechanical formalism

Conservation of probability

  • Self-adjointness of Hamiltonian guarantees unitary time evolution
  • Unitarity preserves norm of state vectors, interpreted as probability conservation
  • Essential self-adjointness ensures unique unitary evolution without need for boundary conditions
  • Crucial for maintaining probabilistic interpretation of quantum mechanics

Time evolution of quantum systems

  • Schrödinger equation involves self-adjoint Hamiltonian: iℏddtψ=Hψi\hbar \frac{d}{dt}ψ = Hψ
  • Essential self-adjointness of H ensures well-defined solution for all times
  • Allows study of long-time behavior and scattering problems
  • Connected to existence and uniqueness of dynamics in classical systems

Measurement theory

  • Self-adjoint operators represent physical observables in quantum mechanics
  • Spectral theorem provides probabilistic interpretation of measurement outcomes
  • Essential self-adjointness ensures unique spectral decomposition
  • Allows consistent definition of expectation values and uncertainty relations

Key Terms to Review (36)

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 Operator: A bounded operator is a linear transformation between two normed vector spaces that maps bounded sets to bounded sets, which implies that there exists a constant such that the operator does not increase the size of vectors beyond a certain limit. This concept is crucial in functional analysis, especially when dealing with operators on Hilbert and Banach spaces, where it relates to various spectral properties and the stability of solutions to differential equations.
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.
Commutator methods: Commutator methods are techniques used in spectral theory to analyze the properties of operators, particularly in relation to their self-adjointness and essential self-adjointness. These methods rely on the commutator of operators to establish conditions under which certain operators can be considered self-adjoint, thus providing insights into their spectra and the behavior of associated differential equations.
Continuous Spectrum: A continuous spectrum refers to the set of values that an operator can take on in a way that forms a continuous interval, rather than discrete points. This concept plays a crucial role in understanding various properties of operators, particularly in distinguishing between bound states and scattering states in quantum mechanics and analyzing the behavior of self-adjoint operators.
David Hilbert: David Hilbert was a prominent German mathematician known for his foundational work in various areas of mathematics, particularly in the field of functional analysis and spectral theory. His contributions laid the groundwork for the modern understanding of Hilbert spaces, which are central to quantum mechanics and spectral theory, connecting concepts such as self-adjoint operators, spectral measures, and the spectral theorem.
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.
Differential Operator: A differential operator is a mathematical operator that involves the differentiation of a function. In spectral theory, differential operators are crucial as they appear in the formulation of differential equations that describe various physical and mathematical phenomena. Understanding their properties, such as self-adjointness and closedness, helps in analyzing the spectral characteristics of linear operators, particularly in relation to boundary value problems and their solutions.
Dirichlet vs Neumann Conditions: Dirichlet and Neumann conditions are boundary conditions used in mathematical physics to specify the behavior of solutions to differential equations at the boundaries of a given domain. Dirichlet conditions require the solution to take specific values on the boundary, while Neumann conditions specify the values of the derivative (usually the normal derivative) on the boundary. These conditions play a crucial role in determining whether an operator is essentially self-adjoint, influencing the existence and uniqueness of solutions to boundary value problems.
Domain of an operator: The domain of an operator refers to the set of input values for which the operator is defined and can produce valid outputs. Understanding the domain is crucial when dealing with self-adjoint operators, their extensions, and issues of essential self-adjointness, as it directly affects the properties and behavior of the operator in mathematical analysis.
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.
Functional Calculus: Functional calculus is a mathematical framework that allows the application of functions to operators, particularly in the context of spectral theory. It provides a way to define new operators using functions applied to existing operators, enabling a deeper analysis of their spectral properties and behaviors. This approach is crucial for understanding how various classes of operators can be manipulated and studied through their spectra.
Hamiltonians on Infinite Domains: Hamiltonians on infinite domains refer to operators that describe the total energy of a system, often represented in quantum mechanics, in spaces that are not confined or limited. These Hamiltonians can exhibit different properties such as essential self-adjointness, which ensures that the operator is uniquely defined and can be associated with a self-adjoint operator, leading to a well-defined spectral theory. This connection is crucial as it affects how solutions to quantum mechanical systems behave under boundary conditions and influences their physical interpretation.
Harmonic oscillator hamiltonian: The harmonic oscillator Hamiltonian is a fundamental operator in quantum mechanics that describes the energy of a quantum harmonic oscillator. It is mathematically represented as $$ rac{p^2}{2m} + rac{1}{2}kx^2$$, where $$p$$ is the momentum operator, $$m$$ is the mass of the particle, $$k$$ is the spring constant, and $$x$$ is the position operator. This Hamiltonian serves as a key example for understanding essential self-adjointness and the properties of quantum systems.
Hilbert space: A Hilbert space is a complete inner product space that provides the framework for many areas in mathematics and physics, particularly in quantum mechanics and functional analysis. It allows for the generalization of concepts such as angles, lengths, and orthogonality to infinite-dimensional spaces, making it essential for understanding various types of operators and their spectral properties.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to various fields, including quantum mechanics, functional analysis, and the foundations of mathematics. His work laid the groundwork for many concepts in spectral theory, particularly regarding self-adjoint operators and their spectra.
Kato-Rellich Theorem: The Kato-Rellich Theorem is a result in spectral theory that provides conditions under which the essential spectrum of a self-adjoint operator remains unchanged under certain perturbations. This theorem is significant in understanding how small changes in operators can affect their eigenvalues and spectra, particularly in the context of unbounded self-adjoint operators and their resolvents.
Krein-von Neumann extension: The Krein-von Neumann extension is a method used to extend a symmetric operator that is not essentially self-adjoint to a self-adjoint operator on a larger Hilbert space. This extension is particularly useful for operators with deficiency indices (m,n) = (1,1), allowing the operator to be treated as self-adjoint in a broader context. The extension provides a way to analyze spectral properties of the original operator by relating them to the extended self-adjoint operator.
Krein's Theorem: Krein's Theorem is a fundamental result in spectral theory that addresses the essential self-adjointness of certain unbounded operators, particularly in the context of differential operators on a Hilbert space. It provides necessary and sufficient conditions under which an operator, which may not be self-adjoint on its domain, can be extended to a self-adjoint operator. This theorem is crucial in understanding the spectral properties of operators, as essential self-adjointness ensures that the operator's spectrum behaves well and guarantees the uniqueness of the self-adjoint extension.
L2 space: l2 space, also known as the space of square-summable sequences, is a Hilbert space that consists of all infinite sequences of complex numbers for which the sum of the squares of the absolute values is finite. This concept is crucial in functional analysis and spectral theory, as it provides a complete and well-defined framework for studying linear operators and their properties, particularly in the context of essential self-adjointness.
Laplace Operator: The Laplace operator, often denoted as $$ abla^2$$ or $$ ext{Δ}$$, is a second-order differential operator that plays a vital role in mathematical analysis, particularly in the study of partial differential equations and spectral theory. It is defined as the divergence of the gradient of a function and is essential in understanding various physical phenomena, such as heat conduction, wave propagation, and vibrations in membranes and plates. This operator connects to important concepts such as self-adjointness and spectral properties of unbounded operators.
Momentum operator: The momentum operator is a fundamental concept in quantum mechanics, typically represented as \\hat{p} = -i\\hbar \\frac{d}{dx} in one dimension, where \\hbar is the reduced Planck's constant and i is the imaginary unit. It acts on wave functions to extract information about a particle's momentum, directly linking quantum mechanics to classical momentum principles through the spectral properties of operators.
Nelson's Analytic Vector Theorem: Nelson's Analytic Vector Theorem states that for a symmetric operator on a Hilbert space, if the operator has a dense domain of analytic vectors, then the operator is essentially self-adjoint. This theorem connects the properties of symmetric operators to essential self-adjointness, providing a key criterion for identifying when a symmetric operator can be extended to a self-adjoint operator.
Point Spectrum: The point spectrum of an operator consists of all the eigenvalues for which there are non-zero eigenvectors. It provides crucial insights into the behavior of operators and their associated functions, connecting to concepts like essential and discrete spectrum, resolvent sets, and various types of operators including self-adjoint and compact ones.
Position Operator: The position operator is a fundamental concept in quantum mechanics, represented by the operator \\hat{x} that acts on the wave functions in a Hilbert space to determine the position of a particle. It plays a crucial role in spectral theory, especially in the context of unbounded self-adjoint operators, where it is used to analyze the spectrum of possible measurement outcomes and understand essential self-adjointness conditions that guarantee the operator's well-defined nature in quantum systems.
Resolvent Set vs Spectrum: The resolvent set consists of all complex numbers for which an operator has a bounded inverse, while the spectrum encompasses all complex numbers that do not belong to the resolvent set. This distinction is crucial in understanding the behavior of operators, particularly in the context of essential self-adjointness, where the nature of the spectrum can reveal important information about the operator's properties and its spectral decomposition.
Robin Boundary Conditions: Robin boundary conditions are a type of boundary condition used in differential equations where the solution is a linear combination of the function itself and its derivative at the boundary. This creates a balance between Dirichlet and Neumann conditions, often modeling physical situations like heat transfer or wave propagation. They play a significant role in defining essential self-adjointness of differential operators, impacting the uniqueness and existence of solutions to boundary value problems.
Schrödinger Operators: Schrödinger operators are a class of differential operators that arise in quantum mechanics, primarily represented as $H = -\frac{d^2}{dx^2} + V(x)$, where $V(x)$ is a potential function. They play a crucial role in studying the behavior of quantum systems and understanding essential self-adjointness, which determines whether a physical observable has a well-defined spectral property.
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 for Self-Adjoint Operators: The spectral theorem for self-adjoint operators states that any self-adjoint operator can be represented in terms of its eigenvalues and eigenvectors, allowing it to be expressed as an integral over a measure associated with a projection-valued measure. This theorem connects the concepts of linear operators on Hilbert spaces to their spectral properties, enabling the decomposition of operators into simpler components, which is crucial for understanding their behavior in various contexts.
Sturm-Liouville Problems: Sturm-Liouville problems refer to a specific type of differential equation problem that involves a linear second-order differential operator and boundary conditions. These problems are significant in mathematical physics as they help in solving various boundary value problems, leading to the understanding of eigenvalues and eigenfunctions, which are crucial in spectral theory and quantum mechanics.
Symmetric operator: A symmetric operator is a linear operator defined on a dense domain in a Hilbert space that satisfies the property \( \langle Ax, y \rangle = \langle x, Ay \rangle \) for all vectors \( x \) and \( y \) in its domain. This means that the operator is equal to its adjoint on that domain. Symmetric operators are important because they can be linked to self-adjointness and the behavior of differential operators, which are crucial in understanding various aspects of quantum mechanics and mathematical physics.
Theorem of von Neumann: The theorem of von Neumann states that a densely defined operator on a Hilbert space is essentially self-adjoint if and only if its closure is self-adjoint. This is significant because it provides a clear criterion for determining whether certain operators can be treated as self-adjoint, which is essential for ensuring the physical and mathematical properties of quantum mechanical systems.
Unbounded Observables: Unbounded observables are operators in quantum mechanics that do not have a finite upper limit on their spectrum, meaning they can take on arbitrarily large values. These observables typically arise in situations where physical quantities are not restricted, such as position or momentum, and require careful mathematical treatment to ensure proper definitions and properties in the context of quantum systems.
Weyl's Limit Point-Limit Circle Criterion: Weyl's Limit Point-Limit Circle Criterion is a method used to determine the self-adjointness of a differential operator based on the behavior of its eigenvalues and eigenfunctions at the boundaries of its domain. This criterion classifies the spectrum of differential operators into two categories: limit point and limit circle, helping to establish whether a given operator can be considered essential self-adjoint. The concept is crucial for understanding the conditions under which a differential operator has a unique self-adjoint extension.
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