9.4 Weyl's theorem

Weyl's theorem is a cornerstone in operator theory, connecting the essential spectrum of self-adjoint operators to their Fredholm properties. It reveals how the spectrum behaves under compact perturbations, offering insights into the stability of physical systems.

This theorem bridges spectral theory and Fredholm theory, showing that the essential spectrum remains unchanged under compact perturbations. It's crucial for analyzing quantum systems, differential operators, and understanding the long-term behavior of physical models.

Weyl's Theorem

Statement and Proof of Weyl's Theorem

• Weyl's theorem asserts that for a self-adjoint operator T on a Hilbert space H, the essential spectrum of T equals the set of all λ in the spectrum of T where T - λI is not a Fredholm operator
• Essential spectrum of an operator T encompasses all points in the spectrum of T that are not isolated eigenvalues of finite multiplicity
• Proof of Weyl's theorem demonstrates the invariance of the essential spectrum under compact perturbations
  • Shows that the difference of resolvents for T and a compact perturbation of T is compact
  • Utilizes the concept of approximate eigenvalues and their relationship to the essential spectrum
• Spectral theory understanding forms the foundation for comprehending Weyl's theorem
  • Includes definitions of spectrum, point spectrum, and continuous spectrum
• Proof steps involve:
  1. Defining the essential spectrum and Fredholm operators
  2. Showing that compact perturbations do not affect the essential spectrum
  3. Demonstrating the equivalence between non-Fredholm points and points in the essential spectrum
• Examples of essential spectra:
  • For the multiplication operator on $L^2([0,1])$, the essential spectrum is the entire range of the multiplier function
  • For the Laplacian on $L^2(\mathbb{R}^n)$, the essential spectrum is $[0,\infty)$

Key Concepts and Definitions

• Self-adjoint operator defined as an operator T such that $T = T^*$ (where $T^*$ is the adjoint of T)
• Fredholm operator characterized by finite-dimensional kernel and cokernel, with closed range
• Compact operator maps bounded sets to relatively compact sets
• Approximate eigenvalue λ of T satisfies $\inf_{x \in H, \|x\|=1} \|(T-λI)x\| = 0$
• Spectrum of an operator T denoted as $\sigma(T)$ includes all λ such that T - λI is not invertible
• Point spectrum consists of eigenvalues, while continuous spectrum contains approximate eigenvalues that are not eigenvalues
• Examples:
  • Identity operator on infinite-dimensional space has essential spectrum {1}
  • Compact operator on infinite-dimensional space has essential spectrum {0}

Applications of Weyl's Theorem

Analysis of Operator Spectra

• Weyl's theorem provides a powerful method for determining the essential spectrum of self-adjoint operators by examining their Fredholm properties
• Application involves identifying the set of λ for which T - λI is not Fredholm, directly corresponding to the essential spectrum
• Differential operators analysis uses Weyl's theorem to examine the continuous spectrum and distinguish it from the point spectrum
• Schrödinger operators utilize Weyl's theorem to determine the stability of the essential spectrum under potential perturbations
• Quantum mechanics applications include studying spectra of Hamiltonians and their perturbations
• Techniques from functional analysis support the application of Weyl's theorem
  • Compact operator theory
  • Spectral theory
• Examples:
  • One-dimensional Schrödinger operator $-\frac{d^2}{dx^2} + V(x)$ on $L^2(\mathbb{R})$ with $V(x) \to 0$ as $|x| \to \infty$ has essential spectrum $[0,\infty)$
  • Sturm-Liouville operator $-\frac{d}{dx}(p(x)\frac{d}{dx}) + q(x)$ on a finite interval with regular endpoints has purely discrete spectrum

Practical Implications in Physics

• Weyl's theorem aids in understanding the stability of physical systems under small perturbations
• Helps predict the behavior of energy levels in quantum systems when subjected to external influences
• Provides insights into the continuous and discrete parts of energy spectra in various physical models
• Assists in analyzing the long-term behavior of quantum systems through spectral properties
• Applications extend to solid-state physics, quantum field theory, and statistical mechanics
• Examples:
  • Hydrogen atom perturbed by a weak electric field (Stark effect)
  • Electrons in a periodic potential (band structure in solids)

Weyl's Theorem and Fredholm Index

Fredholm Index and Essential Spectrum

• Fredholm index of an operator T defined as the difference between the dimension of its kernel and the codimension of its range
• Weyl's theorem establishes a direct link between the essential spectrum and the Fredholm property of operators
• For λ in the essential spectrum, T - λI is not Fredholm
  • Implies either infinite-dimensional kernel or non-closed range
• Connection between Weyl's theorem and Fredholm index provides insights into spectral properties through index theory
• Index theory knowledge required to fully understand this connection and its applications in operator theory
• Stability of the Fredholm index under compact perturbations relates closely to the stability of the essential spectrum in Weyl's theorem
• This relationship forms the foundation for studying spectral flow and its applications in mathematical physics
• Examples:
  • Toeplitz operators on Hardy space have Fredholm index related to the winding number of their symbol
  • Index of elliptic differential operators on compact manifolds given by the Atiyah-Singer index theorem

Applications in Index Theory

• Weyl's theorem helps classify operators based on their Fredholm properties and essential spectra
• Enables the study of topological invariants associated with operators through their Fredholm indices
• Facilitates the analysis of families of operators and their spectral properties
• Provides a framework for understanding the behavior of eigenvalues under continuous deformations of operators
• Applies to the study of boundary value problems and elliptic operators on manifolds
• Contributes to the development of K-theory and its applications in operator algebras
• Examples:
  • Spectral flow of a family of self-adjoint Fredholm operators
  • Index of Dirac operators on spin manifolds

Spectrum Stability under Perturbations

Compact Perturbations and Spectral Stability

• Weyl's theorem implies the stability of the essential spectrum of a self-adjoint operator under compact perturbations
• This stability property enables analysis of perturbed operators by focusing on the essential spectrum of the unperturbed operator
• Provides a method to determine which parts of the spectrum may change under compact perturbations and which remain invariant
• Applications of this stability property prove crucial in quantum mechanics, where compact perturbations often represent physical interactions
• Essential spectrum stability contrasts with the potential instability of the discrete spectrum under compact perturbations
• Understanding this stability requires knowledge of compact operators and their properties in relation to the spectrum of an operator
• Concept of relatively compact perturbations extends the applicability of Weyl's theorem to a broader class of operator perturbations
• Examples:
  • Schrödinger operator with a potential decaying at infinity
  • Sturm-Liouville operator with bounded perturbation

Implications for Operator Theory and Physics

• Weyl's theorem provides a robust framework for studying the effects of perturbations on operator spectra
• Helps in classifying perturbations based on their impact on the essential and discrete spectra
• Guides the development of perturbation theory in quantum mechanics and other areas of mathematical physics
• Assists in understanding the asymptotic behavior of eigenvalues under various types of perturbations
• Contributes to the study of resonances and scattering theory
• Provides insights into the stability of physical systems described by self-adjoint operators
• Examples:
  • Stability of the spectrum of the Laplacian under bounded potential perturbations
  • Analysis of the essential spectrum for operators arising in quantum field theory