7.3 Eigenvalue perturbation

Eigenvalue perturbation is a crucial concept in spectral theory, analyzing how small changes in matrices or operators affect their eigenvalues. This technique provides insights into system behavior under slight variations, making it valuable in quantum mechanics, engineering, and data analysis.

The topic covers various aspects, from basic definitions to advanced applications. It explores first-order and higher-order perturbation theories, matrix perturbation for both simple and degenerate eigenvalues, and analytic perturbation theory. Numerical methods and applications in quantum mechanics are also discussed.

Eigenvalue perturbation basics

• Eigenvalue perturbation analyzes how eigenvalues change when small modifications occur in matrices or operators
• Fundamental concept in spectral theory provides insights into system behavior under slight parameter variations
• Applies to various fields including quantum mechanics, structural engineering, and data analysis

Definition of eigenvalue perturbation

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• Describes changes in eigenvalues resulting from small modifications to a matrix or operator
• Expressed mathematically as $A(\epsilon) = A_0 + \epsilon A_1 + \epsilon^2 A_2 + ...$ where $A_0$ represents the unperturbed matrix
• Aims to find new eigenvalues $\lambda(\epsilon)$ and eigenvectors $v(\epsilon)$ as functions of the perturbation parameter $\epsilon$
• Utilizes series expansions to approximate perturbed eigenvalues and eigenvectors

Importance in spectral theory

• Enables analysis of system stability under small parameter changes
• Provides tools for understanding eigenvalue sensitivity in numerical computations
• Facilitates prediction of system behavior in slightly altered conditions
• Plays crucial role in quantum mechanics for approximating energy levels of perturbed systems
• Aids in understanding degeneracy breaking and level splitting phenomena

Types of perturbations

• Additive perturbations involve adding a small matrix to the original matrix
• Multiplicative perturbations multiply the original matrix by a matrix close to the identity
• Structured perturbations maintain specific matrix properties (symmetry, sparsity)
• Random perturbations introduce stochastic elements to model uncertainty
• Singular perturbations involve changes in the matrix dimension or structure

Perturbation theory fundamentals

• Perturbation theory forms the foundation for analyzing eigenvalue changes in spectral theory
• Provides systematic approach to approximating solutions for complex problems
• Widely used in physics, mathematics, and engineering to study systems under small disturbances

First-order perturbation theory

• Approximates eigenvalue changes using linear terms in the perturbation expansion
• First-order correction to eigenvalue given by $\lambda^{(1)} = v_0^* A_1 v_0$ where $v_0$ represents the unperturbed eigenvector
• Assumes perturbation is small enough for linear approximation to be valid
• Provides quick estimates of eigenvalue shifts for simple perturbations
• Accuracy limited for larger perturbations or near degeneracies

Higher-order perturbation theory

• Extends approximations to include quadratic, cubic, and higher-order terms
• Second-order correction given by $\lambda^{(2)} = \sum_{k \neq 0} \frac{|(v_k^* A_1 v_0)|^2}{\lambda_0 - \lambda_k}$ where $\lambda_k$ and $v_k$ represent unperturbed eigenvalues and eigenvectors
• Improves accuracy for larger perturbations or near-degenerate cases
• Requires more computational effort compared to first-order theory
• Convergence may become an issue for very high-order expansions

Convergence of perturbation series

• Analyzes whether the infinite series of perturbation terms converges to the true solution
• Depends on the nature of the perturbation and the spectral properties of the unperturbed operator
• Radius of convergence determined by the distance to the nearest singularity in the complex plane
• Asymptotic series may provide useful approximations even when not convergent
• Techniques like Padé approximants can improve convergence in some cases

Matrix perturbation theory

• Focuses on eigenvalue and eigenvector changes in finite-dimensional matrices
• Provides tools for analyzing stability and sensitivity of matrix eigenvalue problems
• Applies to both Hermitian and non-Hermitian matrices with distinct considerations

Perturbation of simple eigenvalues

• Addresses changes in non-degenerate eigenvalues under small matrix perturbations
• First-order approximation given by $\lambda(\epsilon) \approx \lambda_0 + \epsilon \frac{u_0^* A_1 v_0}{u_0^* v_0}$ where $u_0$ and $v_0$ are left and right eigenvectors
• Eigenvector perturbation expressed as linear combination of unperturbed eigenvectors
• Bauer-Fike theorem provides bounds on eigenvalue perturbations
• Condition number of eigenvector influences sensitivity to perturbations

Perturbation of degenerate eigenvalues

• Analyzes changes in eigenvalues with algebraic multiplicity greater than one
• Perturbation typically lifts degeneracy, splitting eigenvalues into distinct values
• Requires degenerate perturbation theory, often using reduced resolvent techniques
• First-order approximation involves solving a characteristic equation within the degenerate subspace
• May lead to non-analytic behavior in eigenvalue and eigenvector perturbation expansions

Left and right eigenvectors

• Distinguishes between left ($u^*A = \lambda u^*$) and right ($Av = \lambda v$) eigenvectors for non-Hermitian matrices
• Left and right eigenvectors coincide for Hermitian matrices
• Biorthogonality relation $u_i^* v_j = 0$ for $i \neq j$ useful in perturbation calculations
• Normalization convention $u_i^* v_i = 1$ simplifies perturbation formulas
• Sensitivity of eigenvalues related to the angle between left and right eigenvectors

Analytic perturbation theory

• Examines behavior of eigenvalues and eigenvectors as analytic functions of perturbation parameter
• Provides rigorous mathematical framework for understanding perturbation effects
• Applies complex analysis techniques to study eigenvalue problems

Kato-Rellich theorem

• Establishes conditions for analytic dependence of simple eigenvalues on perturbation parameter
• States that for analytic family of operators, simple eigenvalues and associated eigenprojections are also analytic
• Requires that perturbation does not cause eigenvalue crossings
• Provides basis for power series expansions of eigenvalues and eigenvectors
• Extends to more general cases with appropriate modifications

Analytic continuation of eigenvalues

• Explores how eigenvalues behave when perturbation parameter extended to complex plane
• Allows tracking of eigenvalues through parameter space, revealing global structure
• May encounter branch points where eigenvalues exchange roles
• Riemann surface structure emerges for multi-valued eigenvalue functions
• Useful for understanding resonances and exceptional points in non-Hermitian systems

Puiseux series expansions

• Generalizes power series expansions for eigenvalues near branch points
• Takes form $\lambda(\epsilon) = \lambda_0 + c_1 \epsilon^{1/n} + c_2 \epsilon^{2/n} + ...$ where $n$ represents branching order
• Describes behavior of eigenvalues in vicinity of exceptional points
• Provides tool for analyzing non-analytic perturbations
• Connects to algebraic geometry through study of algebraic curves

Numerical methods

• Focuses on computational techniques for solving eigenvalue perturbation problems
• Provides practical tools for analyzing large-scale systems and complex perturbations
• Balances accuracy, efficiency, and numerical stability in calculations

Rayleigh-Schrödinger perturbation theory

• Systematic approach for computing perturbation expansions to arbitrary order
• Recursive formulas for calculating higher-order corrections to eigenvalues and eigenvectors
• Particularly useful in quantum mechanics for energy level calculations
• Can be implemented efficiently using symbolic computation or automatic differentiation
• May suffer from slow convergence or divergence for strong perturbations

Gerschgorin circle theorem

• Provides bounds on eigenvalue locations based on matrix entries
• States that all eigenvalues lie within union of circles centered at diagonal entries
• Circle radii determined by sum of absolute values of off-diagonal entries in each row or column
• Useful for quick estimates of eigenvalue perturbations without full diagonalization
• Can be refined using similarity transformations or iterative techniques

Bauer-Fike theorem

• Establishes bounds on eigenvalue perturbations for diagonalizable matrices
• States that perturbed eigenvalues lie within circles centered at unperturbed eigenvalues
• Circle radii proportional to perturbation norm and condition number of eigenvector matrix
• Provides sharper bounds compared to Gerschgorin theorem for well-conditioned problems
• Generalizes to non-diagonalizable cases using Jordan canonical form

Applications in quantum mechanics

• Eigenvalue perturbation theory plays crucial role in understanding quantum systems
• Allows approximation of energy levels and wavefunctions for complex Hamiltonians
• Provides insights into spectral properties of atoms and molecules under external fields

Time-independent perturbation theory

• Addresses stationary states of quantum systems under constant perturbations
• Expands Hamiltonian as $H = H_0 + \lambda V$ where $H_0$ represents unperturbed system and $V$ perturbation
• Derives corrections to energy levels and wavefunctions as power series in $\lambda$
• Widely used for calculating atomic and molecular properties (polarizabilities, hyperfine structure)
• Requires careful treatment of degenerate states using degenerate perturbation theory

Stark effect

• Describes splitting and shifting of spectral lines in presence of external electric field
• First-order Stark effect vanishes for atoms with definite parity (hydrogen exception)
• Quadratic Stark effect dominates for most atoms, proportional to square of electric field strength
• Perturbation Hamiltonian given by $V = -\mathbf{d} \cdot \mathbf{E}$ where $\mathbf{d}$ represents electric dipole moment
• Leads to mixing of states with different parity, allowing forbidden transitions

Zeeman effect

• Analyzes splitting of energy levels in presence of external magnetic field
• Normal Zeeman effect occurs for singlet states, with equidistant energy level splitting
• Anomalous Zeeman effect arises for multiplet states due to spin-orbit coupling
• Perturbation Hamiltonian given by $V = \mu_B \mathbf{B} \cdot (\mathbf{L} + g_s\mathbf{S})$ where $\mu_B$ represents Bohr magneton
• Paschen-Back effect occurs for strong magnetic fields, decoupling spin and orbital angular momenta

Stability analysis

• Examines sensitivity of eigenvalues to perturbations in matrix or operator
• Crucial for understanding robustness of numerical algorithms and physical systems
• Provides tools for assessing reliability of eigenvalue computations

Eigenvalue sensitivity

• Measures how much eigenvalues change under small perturbations to matrix entries
• First-order sensitivity given by left and right eigenvectors: $\frac{d\lambda}{da_{ij}} = u_i^* v_j$
• Higher sensitivities indicate less stable eigenvalues under perturbations
• Can be visualized using eigenvalue condition numbers or pseudospectra
• Important for identifying potentially problematic eigenvalues in numerical computations

Condition numbers

• Quantify worst-case sensitivity of eigenvalues to perturbations
• Eigenvalue condition number defined as $\kappa(\lambda) = \frac{||u|| \cdot ||v||}{|u^*v|}$ for left and right eigenvectors $u$ and $v$
• Large condition numbers indicate ill-conditioned eigenvalues, highly sensitive to perturbations
• Related to angle between left and right eigenvectors in non-Hermitian case
• Useful for assessing reliability of computed eigenvalues and guiding numerical algorithm choices

Pseudospectra

• Generalizes concept of spectrum to include near-eigenvalues under perturbations
• $\epsilon$-pseudospectrum defined as set of complex numbers $z$ such that $||(A-zI)^{-1}|| \geq \epsilon^{-1}$
• Provides visual representation of eigenvalue sensitivity in complex plane
• Large pseudospectral sets indicate high sensitivity to perturbations
• Useful for analyzing non-normal matrices and operators in fluid dynamics and control theory

Perturbation of generalized eigenvalue problems

• Addresses eigenvalue perturbations in problems of form $Ax = \lambda Bx$
• Arises in various applications including structural dynamics and discretized PDEs
• Requires special considerations due to potential singularity of $B$ matrix

Definite vs indefinite pencils

• Definite pencils have form $A - \lambda B$ where $A$ and $B$ are Hermitian and $B$ positive definite
• Indefinite pencils allow for indefinite $B$ or non-Hermitian $A$
• Definite pencils guarantee real eigenvalues and orthogonality properties
• Indefinite pencils may have complex eigenvalues and require more careful analysis
• Perturbation theory differs significantly between definite and indefinite cases

Perturbation bounds

• Establishes limits on eigenvalue changes for perturbed generalized eigenvalue problems
• Crawford number plays role analogous to smallest singular value in standard eigenvalue problems
• Bounds often involve generalized condition numbers incorporating both $A$ and $B$ matrices
• May require simultaneous perturbations of $A$ and $B$ for meaningful results
• Special techniques needed for infinite or zero eigenvalues

Relative perturbation theory

• Focuses on perturbations relative to magnitude of matrix entries rather than absolute changes
• Particularly useful for problems with widely varying scales in matrix entries
• Develops bounds and expansions in terms of relative changes to $A$ and $B$
• Often provides sharper results compared to classical absolute perturbation theory
• Requires careful scaling and balancing of matrices for optimal results

Advanced topics

• Explores cutting-edge areas of research in eigenvalue perturbation theory
• Addresses complex problems arising in modern applications of spectral theory
• Combines techniques from multiple mathematical disciplines

Non-linear eigenvalue problems

• Studies eigenvalue problems where parameter appears non-linearly in characteristic equation
• Arises in areas such as vibration analysis of structures with frequency-dependent properties
• Perturbation theory must account for non-linear dependence on eigenvalue parameter
• May exhibit more complex branching behavior compared to linear problems
• Requires specialized numerical methods like contour integration or Newton-type iterations

Multiparameter spectral theory

• Examines eigenvalue problems depending on multiple parameters simultaneously
• Generalizes perturbation theory to multi-dimensional parameter spaces
• Applications include quantum systems with multiple coupling constants
• Analyzes singularities and bifurcations in higher-dimensional parameter spaces
• Connects to algebraic geometry through study of discriminant varieties

Infinite-dimensional perturbation theory

• Extends perturbation analysis to operators on infinite-dimensional Hilbert spaces
• Addresses challenges of unbounded operators and continuous spectra
• Requires careful treatment of domain issues and spectral theory of self-adjoint operators
• Applications in quantum field theory and partial differential equations
• Connects to functional analysis and spectral theory of unbounded operators