Glossary

7.2 Functional calculus for unbounded self-adjoint operators

4 min read•august 16, 2024

Functional calculus for unbounded self-adjoint operators lets us apply functions to operators that aren't defined everywhere. It's like a supercharged version of plugging numbers into functions, but for operators instead.

This topic builds on the we learned earlier. It shows how to create new operators by applying functions to unbounded self-adjoint ones, opening up a world of possibilities for analyzing complex systems.

Functional Calculus for Unbounded Operators

Extending Function Application to Unbounded Operators

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  • Functional calculus for unbounded self-adjoint operators broadens the concept of applying functions to operators beyond the bounded case
  • Unbounded self-adjoint operators are densely defined linear operators on a Hilbert space equal to their adjoint but may not be defined on the entire space
  • Spectral theorem for unbounded self-adjoint operators forms the foundation for defining the functional calculus
  • Associates a Borel measurable function f with an operator f(A), where A is an
  • of f(A) comprises vectors ψ in the Hilbert space where the integral of |f(λ)|² with respect to the spectral measure is finite
  • Action of f(A) on a vector ψ in its domain involves integrating f(λ) against the spectral measure of A applied to ψ
  • Preserves algebraic operations allowing manipulation of unbounded self-adjoint operators through functions

Mathematical Framework and Domain Considerations

  • Spectral measure of unbounded self- A acts as a projection-valued measure on Borel subsets of the real line
  • Borel measurable function f applies to operator A
  • Domain of f(A) determined by vectors ψ where integral of |f(λ)|² with respect to spectral measure of A is finite
  • Spectral integral defines f(A)'s action on vectors in its domain: f(A)ψ=∫f(λ)dEA(λ)ψf(A)ψ = ∫ f(λ) dE_A(λ)ψ
    • E_A represents the spectral measure of A
  • Constructed operator f(A) must be well-defined and closed
  • Polynomial or rational functions express f(A) in terms of powers of A and its
  • Complex functions approximate f(A) using simpler functions or limiting procedures when necessary

Constructing Functions of Unbounded Operators

Spectral Measure and Function Selection

  • Identify spectral measure of unbounded self-adjoint operator A (projection-valued measure on Borel subsets of real line)
  • Choose appropriate Borel measurable function f to apply to operator A
  • Determine domain of f(A) by considering vectors ψ where integral of |f(λ)|² with respect to spectral measure is finite
  • Construct f(A) by defining its action on vectors in its domain using spectral integral
  • Verify constructed operator f(A) is well-defined and closed
  • Express f(A) in terms of powers of A and its resolvent for polynomial or rational functions (x2+3xx^2 + 3x, 1x−2\frac{1}{x-2})
  • Approximate f(A) using simpler functions or limiting procedures for complex functions (exponential, logarithmic)

Examples of Function Construction

  • Construct square root of positive self-adjoint operator A: A=∫0∞λdEA(λ)\sqrt{A} = \int_0^∞ \sqrt{λ} dE_A(λ)
  • Define exponential of self-adjoint operator A: eA=∫−∞∞eλdEA(λ)e^A = \int_{-∞}^∞ e^λ dE_A(λ)
  • Create resolvent of self-adjoint operator A: (A−zI)−1=∫−∞∞1λ−zdEA(λ)(A - zI)^{-1} = \int_{-∞}^∞ \frac{1}{λ - z} dE_A(λ), for z not in spectrum of A
  • Form projection onto spectral subspace: P[a,b]=∫abdEA(λ)P_{[a,b]} = \int_a^b dE_A(λ) for interval [a,b]

Properties of the Functional Calculus

Algebraic and Spectral Properties

  • Functional calculus acts as a *-homomorphism preserving algebraic and adjoint operations
  • Demonstrates linearity: (αf+βg)(A)=αf(A)+βg(A)(αf + βg)(A) = αf(A) + βg(A) for scalar α and β, and functions f and g
  • Establishes multiplicative property: (fg)(A)=f(A)g(A)(fg)(A) = f(A)g(A) for suitable functions f and g
  • Respects operator ordering if f ≤ g pointwise, then f(A) ≤ g(A) in operator sense
  • Proves spectral mapping theorem: σ(f(A))=f(σ(A))σ(f(A)) = f(σ(A)), where σ denotes the spectrum
  • Demonstrates continuity properties with respect to different topologies on function spaces
  • Establishes relationship between domain of f(A) and growth properties of function f

Proofs and Theoretical Foundations

  • Prove linearity using properties of spectral integral and
  • Establish multiplicative property through careful analysis of domains and spectral measures
  • Demonstrate operator ordering preservation using functional calculus definition and spectral theorem
  • Prove spectral mapping theorem by analyzing resolvent and spectrum of f(A)
  • Show continuity properties using convergence theorems for spectral integrals
  • Establish domain relationships by examining growth conditions on functions and spectral measures

Applications of the Functional Calculus

Operator Analysis and Quantum Mechanics

  • Define and analyze square root, exponential, and elementary functions of unbounded self-adjoint operators
  • Study heat equation and partial differential equations involving self-adjoint operators
  • Investigate fractional powers of positive self-adjoint operators (A1/3A^{1/3}, A−1/2A^{-1/2})
  • Analyze spectral properties of unbounded self-adjoint operators (continuous and )
  • Define and study semigroups generated by unbounded self-adjoint operators
  • Investigate absolutely continuous and singular parts of spectrum of unbounded self-adjoint operators
  • Define observables and study their properties in infinite-dimensional Hilbert spaces for quantum mechanics

Practical Examples and Problem-Solving

  • Apply functional calculus to Laplacian operator to solve heat equation: u(t)=e−tΔu0u(t) = e^{-tΔ}u_0
  • Use functional calculus to define momentum operator in quantum mechanics: P=−iℏddxP = -i\hbar \frac{d}{dx}
  • Analyze hydrogen atom Hamiltonian using functional calculus: H=−ℏ22mΔ−e2rH = -\frac{\hbar^2}{2m}\Delta - \frac{e^2}{r}
  • Study vibrating string equation through functional calculus of Laplace operator
  • Investigate fractional diffusion equations using fractional powers of Laplacian
  • Apply functional calculus to study quantum harmonic oscillator: H=12mP2+12mω2X2H = \frac{1}{2m}P^2 + \frac{1}{2}mω^2X^2

Key Terms to Review (18)

Adjoint Operator: An adjoint operator is a linear operator associated with a given linear operator, where the action of the adjoint operator relates to an inner product in such a way that it preserves certain properties. The adjoint is crucial for understanding the relationship between operators, especially in the context of functional analysis, where it helps analyze boundedness and self-adjointness of operators.
Analytic function: An analytic function is a complex function that is locally given by a convergent power series. This means that around any point in its domain, the function can be expressed as a sum of powers of the variable, indicating that it is infinitely differentiable. Analytic functions have many important properties, such as being conformal and satisfying the Cauchy-Riemann equations, which make them vital in various areas, including functional analysis and operator theory.
Borel Function: A Borel function is a function that is measurable with respect to the Borel sigma-algebra, which consists of all open sets in a given topological space. These functions play an important role in analysis and probability theory, particularly when dealing with real-valued functions on the real line, as they allow for the extension of certain mathematical operations and provide a framework for defining integrals and limits in more complex settings.
Borel functional calculus: Borel functional calculus is a mathematical framework that allows for the application of Borel-measurable functions to self-adjoint operators on a Hilbert space. This approach extends the notion of applying functions to operators beyond polynomials and rational functions, enabling a broader range of functions to be used in spectral theory. By utilizing the Borel set theory, this calculus provides powerful tools for analyzing unbounded self-adjoint operators, particularly in relation to their spectra and functional properties.
Bounded operator: A bounded operator is a linear transformation between two normed vector spaces that maps bounded sets to bounded sets, meaning it has a finite operator norm. This concept is crucial as it ensures the continuity of the operator, which is connected to convergence and stability in various mathematical contexts, impacting the spectrum of the operator and its behavior in functional analysis.
Continuous Functional: A continuous functional is a linear functional on a topological vector space that maps convergent sequences to convergent sequences, preserving the topology of the space. This means that if a sequence of vectors converges to a limit in the space, the functional applied to this sequence will converge to the functional applied to the limit. In the context of unbounded self-adjoint operators, continuous functionals play a key role in the development of the functional calculus, where they help extend operations on operators to broader settings.
Continuous Spectrum: The continuous spectrum refers to the set of values (often real numbers) that an operator can take on, without any gaps, particularly in relation to its eigenvalues. This concept is crucial in distinguishing between different types of spectra, such as point and residual spectra, and plays a key role in understanding various properties of operators.
David Hilbert: David Hilbert was a prominent German mathematician who made significant contributions to various fields of mathematics, particularly in the areas of functional analysis and operator theory. His work laid the foundational principles for understanding infinite-dimensional spaces and self-adjoint operators, which are crucial in modern mathematical physics and analysis.
Domain: In the context of operator theory, the domain of an operator refers to the set of elements for which the operator is defined and can be applied. Understanding the domain is crucial because it determines where the operator behaves in a well-defined manner, especially when dealing with unbounded linear operators, as they can have more complex and nuanced behaviors compared to bounded operators.
Functional Calculus Representation: Functional calculus representation is a mathematical framework that extends the concept of applying functions to operators, particularly unbounded self-adjoint operators, allowing for the manipulation and evaluation of functions on these operators. This representation is essential for understanding how different functions can interact with operators in Hilbert spaces, enabling various applications in quantum mechanics and differential equations. It provides a way to link operator theory to broader functional analysis by defining how functions can be interpreted as operators.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to various fields, including operator theory, quantum mechanics, and game theory. His work laid the foundation for much of modern mathematics and theoretical physics, particularly in the context of functional analysis and the mathematical formulation of quantum mechanics.
Measure Theory: Measure theory is a branch of mathematics that deals with the systematic way of assigning a number (a measure) to subsets of a given space, allowing for the rigorous development of concepts such as length, area, volume, and probability. This foundational framework enables the exploration of functions and their properties, particularly in defining integrals and limits in more complex spaces. It's crucial for understanding unbounded self-adjoint operators and their functional calculus as it provides the necessary tools to handle functions defined on the spectrum of these operators.
Operator-valued measures: Operator-valued measures are mathematical functions that assign to each measurable set an operator on a Hilbert space, typically in the context of functional analysis. These measures extend the classical notion of measures by incorporating operators, enabling a richer structure for dealing with unbounded self-adjoint operators. They play a crucial role in functional calculus, allowing us to define spectral measures that help analyze the behavior of these operators through integration.
Point Spectrum: The point spectrum of an operator consists of the set of eigenvalues for that operator, specifically those values for which the operator does not have a bounded inverse. These eigenvalues are significant as they correspond to vectors in the Hilbert space that are annihilated by the operator minus the eigenvalue times the identity operator.
Resolvent: The resolvent of an operator is a powerful tool that helps us understand the operator's behavior by relating it to complex numbers. Specifically, for a bounded linear operator $T$, the resolvent is defined as the operator $(T - au I)^{-1}$, where $ au$ is a complex number not in the spectrum of $T$. This relationship allows us to analyze operators' properties and facilitates functional calculus and spectral theory.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra and functional analysis that characterizes self-adjoint operators on Hilbert spaces, providing a way to diagonalize these operators in terms of their eigenvalues and eigenvectors. It connects various concepts such as eigenvalues, adjoint operators, and the spectral properties of bounded and unbounded operators, making it essential for understanding many areas in mathematics and physics.
Stone's Theorem: Stone's Theorem is a fundamental result in functional analysis that provides a framework for understanding the spectral properties of self-adjoint operators through functional calculus. It essentially states that any bounded self-adjoint operator can be represented via continuous functions on its spectrum, allowing us to extend the notion of functions acting on operators. This theorem is crucial for dealing with both bounded and unbounded self-adjoint operators, especially when considering their spectral characteristics.
Unbounded Self-Adjoint Operator: An unbounded self-adjoint operator is a linear operator defined on a Hilbert space that is not bounded, meaning it does not have a finite operator norm, but is still equal to its adjoint. These operators are crucial in quantum mechanics and functional analysis, where they often represent physical observables. Understanding these operators is key to applying the spectral theorem and developing a functional calculus for them.
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