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Mathematical function of a linear operator
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that
Eigenfunction
Mathematical model which is both linear and time-invariant
to multiplication in the frequency domain. For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials
Linear_time-invariant_system
Quantum state with all observables independent of time
energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital
Stationary_state
Construction for adding objects to a Hilbert space
function such as x ↦ e i x , {\displaystyle x\mapsto e^{ix},} is an eigenfunction of the differential operator − i d d x {\displaystyle -i{\frac {d}{dx}}}
Rigged_Hilbert_space
Branch of mathematical analysis
{z^{k}}{\Gamma (\alpha k+1)}},\qquad z\in \mathbb {C} ,} satisfies the eigenfunction equation 0 C D t α E α ( μ t α ) = μ E α ( μ t α ) . {\displaystyle
Fractional_calculus
British mathematician
Heath-Brown (1986) Eigenfunction Expansions Associated with Second-order Differential Equations. Part I (1946) 2nd. edition (1962); Eigenfunction Expansions Associated
Edward_Charles_Titchmarsh
Concepts from linear algebra
{\tfrac {d}{dx}}} , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as d d x e λ x =
Eigenvalues_and_eigenvectors
Electromagnetic quantum-mechanical effect in regions of zero magnetic and electric field
should be no effect. From the Schrödinger equation, the phase of an eigenfunction with energy E {\displaystyle E} goes as e − i E t / ℏ {\displaystyle
Aharonov–Bohm_effect
Statistical method for investigating the dominant modes of variation of functional data
is an orthonormal basis of the Hilbert space L2 that consists of the eigenfunctions of the autocovariance operator. FPCA represents functional data in the
Functional principal component analysis
Functional_principal_component_analysis
mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint
Hilbert–Schmidt_theorem
Irreducible representation of the rotation group SO
theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter
Wigner_D-matrix
Any sheaf whose value is based on an eigenfunction
mathematics, a Hecke eigensheaf is any sheaf whose value is based on an eigenfunction. It is an object that is a tensor-multiple of itself when formed under
Hecke_eigensheaf
Class of ordinary differential equations
y=y(x)} of the problem. Such functions y {\displaystyle y} are called the eigenfunctions associated to each λ {\displaystyle \lambda } . Sturm–Liouville theory
Sturm–Liouville_theory
Mathematical transform that expresses a function of time as a function of frequency
transform is a linear transform that has eigenfunctions obeying ℱ[ψ]=λψ, with λ∈ℂ. A set of eigenfunctions is found by noting that the homogeneous
Fourier_transform
Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes
Spectral_shape_analysis
Russian mathematician (born 1989)
estimate (from above) for Hausdorff measures on the zero sets of Laplace eigenfunctions defined on compact smooth manifolds and an estimate (from below) in
Aleksandr Logunov (mathematician)
Aleksandr_Logunov_(mathematician)
Area of mathematics
possible modes of vibration of a circular membrane. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional
Functional_analysis
Function in harmonic analysis on groups
functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra
Zonal_spherical_function
Number, approximately 3.14
The overtones of a vibrating string are eigenfunctions of the second derivative, and form a harmonic progression. The associated eigenvalues form the arithmetic
Pi
Theorem in quantum mechanics
ψ λ ⟩ {\displaystyle |\psi _{\lambda }\rangle } , is an eigenstate (eigenfunction) of the Hamiltonian, depending implicitly upon λ , {\displaystyle \lambda
Hellmann–Feynman_theorem
Q-analog in combinatorial mathematics
q-exponential is a q-analog of the exponential function, namely the eigenfunction of a q-derivative. There are many q-derivatives, for example, the classical
Q-exponential
potential system, if a wavefunction ψ ( r ) {\displaystyle \psi (r)} is an eigenfunction of the Hamiltonian operator H ^ ( p ^ , x ^ ) {\displaystyle {\hat {H}}({\hat
Inverse_square_potential
Type of problem involving ODEs or PDEs
problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary
Boundary_value_problem
Operator generalizing the Laplacian in differential geometry
{\displaystyle -\Delta u=\lambda u,} where u {\displaystyle u} is the eigenfunction associated with the eigenvalue λ {\displaystyle \lambda } . It can be
Laplace–Beltrami_operator
Principle of quantum mechanics
the state of a system is given by a linear combination of all the eigenfunctions of the Schrödinger equation governing that system. An example is a qubit
Quantum_superposition
Theory of NMR spectroscopy based on Quantum mechanics
Spin states Eigenfunction Eigenvalue (energy) αα ψα,1 ψα,2 +(1/2)v0,1 + (1/2)v0,2 αβ ψα,1 ψβ,2 +(1/2)v0,1 - (1/2)v0,2 βα ψβ,1 ψα,2 -(1/2)v0,1 + (1/2)v0
Quantum mechanics of nuclear magnetic resonance spectroscopy
Quantum_mechanics_of_nuclear_magnetic_resonance_spectroscopy
Type of vector space in math
where K is a continuous function symmetric in x and y. The resulting eigenfunction expansion expresses the function K as a series of the form K ( x , y
Hilbert_space
Non-self-adjoint compact operator used to solve boundary value problems for the Laplacian
function gives an eigenfunction with eigenvalue 1/2 and multiplicity one; that there are no corresponding generalized eigenfunctions with eigenvalue 1/2;
Neumann–Poincaré_operator
Probability distribution
variance 1/σ2. In particular, the standard normal distribution φ is an eigenfunction of the Fourier transform. In probability theory, the Fourier transform
Normal_distribution
Part of spectral theory
spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
Function acting on the space of physical states in physics
(such as position, momentum, energy, angular momentum etc.). If ψ is an eigenfunction of the operator A ^ {\displaystyle {\hat {A}}} , then A ^ ψ = a ψ ,
Operator_(physics)
Quantum mechanical operator related to rotational symmetry
quantum problems involving rotational symmetry. Being an observable, its eigenfunctions represent the distinguishable physical states of a system's angular
Angular_momentum_operator
Method for approximating eigenvalues
Hamiltonian, it uses trial wave functions to approximate the ground-state eigenfunction. In the context of the finite-element method, it is mathematically the
Rayleigh–Ritz_method
of spin contamination are undesirable. In particular, they are not eigenfunctions of the total spin-squared operator, Ŝ2, but can formally be expanded
Spin_contamination
Differential operator in mathematics
spectrum, and its eigenfunctions form an orthonormal basis of L 2 ( M ) {\displaystyle L^{2}(M)} . On the round sphere, these eigenfunctions are the spherical
Laplace_operator
Description of a quantum-mechanical system
equation is an eigenvalue equation. Therefore, the wave function is an eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) E {\displaystyle
Schrödinger_equation
Polynomial sequence
\varphi ),\qquad k=0,\pm 1,\pm 2,\cdots .} The Zernike polynomials are eigenfunctions of the Zernike differential operator, in modern formulation L [ f ]
Zernike_polynomials
Type of mathematical model
translational invariance ψ = ∂x u0(x) is a neutral eigenfunction with the eigenvalue λ = 0, and all other eigenfunctions can be sorted according to an increasing
Reaction–diffusion_system
Quantum Monte Carlo method that uses a Green's function
not a simple number or function. There are special functions, called eigenfunctions, for which HΨ(x)=EΨ(x), where E is a number. These functions are special
Diffusion_Monte_Carlo
Equation used in quantum scattering problems
which the two systems are infinitely far apart and do not interact. Its eigenfunctions are | ϕ ⟩ {\displaystyle |\phi \rangle \,} and its eigenvalues are the
Lippmann–Schwinger_equation
ergodic in the sense that the probability density associated to the nth eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit
Quantum_ergodicity
Approximation method in quantum physics
same radial part and to restrict the variational solution to be a spin eigenfunction. Even so, calculating a solution by hand using the Hartree–Fock equations
Hartree–Fock_method
Special type of functions in mathematics
In mathematics, prolate spheroidal wave functions (PSWFs) are eigenfunctions of the Laplacian in prolate spheroidal coordinates, adapted to boundary conditions
Prolate spheroidal wave function
Prolate_spheroidal_wave_function
p(x) = x, which has eigenvalues n = 0, 1, 2, 3, ... and corresponding eigenfunctions xn. Cauchy–Euler equation Sturm–Liouville theory Ross, Clay C (2004)
Cauchy–Euler_operator
American mathematician (1953–2022)
the asymptotic and distribution of its eigenfunctions (e.g. quantum ergodicity, equidistribution of eigenfunctions in billiard geometries, quantum ergodic
Steven_Zelditch
Representation theory
groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated
Plancherel theorem for spherical functions
Plancherel_theorem_for_spherical_functions
Any entity that can be measured
incompatible. Incompatible observables cannot have a complete set of common eigenfunctions. Note that there can be some simultaneous eigenvectors of A ^ {\displaystyle
Observable
Indian mathematician (1913–1968)
1949, the two wrote a paper together called, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, in which they introduced
Subbaramiah_Minakshisundaram
Mathematical functions and constants
uniform grid. These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find
Eigenvalues and eigenvectors of the second derivative
Eigenvalues_and_eigenvectors_of_the_second_derivative
Coefficients in angular momentum eigenstates of quantum systems
can be read directly from this approach as spherical harmonics are eigenfunctions of total angular momentum and projection thereof onto an axis, and the
Clebsch–Gordan_coefficients
the character. The Walsh functions ψ k {\displaystyle \psi _{k}} are eigenfunctions of the dyadic differentiation operator with corresponding eigenvalues
Dyadic_derivative
Study of classical optics using Fourier transforms
eigenfunction solutions / eigenvector solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions /
Fourier_optics
Operators useful in quantum mechanics
}{2}}\psi _{0}=E_{0}\psi _{0}.} So ψ 0 {\displaystyle \psi _{0}} is an eigenfunction of the Hamiltonian. This gives the ground state energy E 0 = ℏ ω / 2
Creation and annihilation operators
Creation_and_annihilation_operators
Sum of a product and its opposite equals zero.
term antisymmetrical is commonly used in physics. "An antisymmetrical eigenfunction vanishes identically when two of the electrons are in the same orbit
Anticommutative_property
Israeli mathematician (1922–2025)
differential equations include Agmon's method for proving exponential decay of eigenfunctions for elliptic operators. In 1965 he published a book on linear boundary
Shmuel_Agmon
American chemist (born 1937)
doctoral studies, Goddard developed a many-electron method built on eigenfunctions of total spin, published in a 1967 series in Physical Review as an improved
William_Andrew_Goddard_III
Symmetry of spatially mirrored systems
is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of H ^ {\displaystyle {\hat {H}}} is either invariant to P ^ {\displaystyle
Parity_(physics)
Model for the potential energy of a diatomic molecule
(grey) and Morse (black) potential curves are shown along with their eigenfunctions (respectively green and blue for harmonic oscillator and Morse) for
Morse_potential
Eigenvalue transformation method
negative eigenvalue of the Schrödinger operator with corresponding eigenfunction ψ ∈ L 2 ( R n ) {\displaystyle \psi \in L^{2}(\mathbb {R} ^{n})} , then
Birman–Schwinger_principle
Aspect of mathematical spectrum theory
real line. Physicists say that each k∈ℝ is an eigenvalue of T with eigenfunction eikx. However, this is not technically correct, since eikx has infinite
Essential_spectrum
Quantum mechanics concept
ψ {\displaystyle \psi } is the (complex valued) wavefunction, or "eigenfunction", and E {\displaystyle E} is the energy, a real number, sometimes called
Finite_potential_well
Discrete-time dynamical system
analysis is to find the eigenfunctions φk and eigenvalues λk of this operator, which satisfy Uφk = λkφk. These eigenfunctions, also known as Koopman modes
Hénon_map
Physical spaces representing position and momentum, Fourier-transform duals
long as they span the state space. If one chooses the (generalized) eigenfunctions of the position operator as a set of basis functions, one speaks of
Position_and_momentum_spaces
Differential equation for the description of waves or standing wave
expanding f and g in the eigenfunctions of the Laplacian in D, which satisfy the boundary conditions. Thus the eigenfunction v satisfies ∇ ⋅ ∇ v + λ v
Wave_equation
Transition rate formula
tE_{n}/\hbar }=0,} where En and |n⟩ are the stationary eigenvalues and eigenfunctions of H0. This equation can be rewritten as a system of differential equations
Fermi's_golden_rule
Stochastic process modeling random walk with friction
106. Holmes-Cerfon, Miranda (2022). "Lecture 12: Detailed balance and Eigenfunction methods" (PDF). Aït-Sahalia 2002, pp. 223–262. Kloeden, Platen & Schurz
Ornstein–Uhlenbeck_process
Operator on a Hilbert space that shifts basis vectors
of square-integrable functions. When acting on a measure space, the eigenfunctions of shift operators are characteristically fractal in shape, often
Unilateral_shift_operator
Non-convergence in Fourier analysis
space and other non-compact rank-one symmetric spaces. Also related are eigenfunction expansions on a geodesic ball in a rank-one symmetric space, but one
Pinsky_phenomenon
Chaotic map from the unit square into itself
map is an exactly solvable model of deterministic chaos, in that the eigenfunctions and eigenvalues of the transfer operator can be explicitly determined
Baker's_map
Mathematical function describing fluid motion
In applied mathematics, the Hough functions are the eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such
Hough_function
Periodic vortex array in fluid dynamics
ω=λψ, where λ is a constant. Then ψ must satisfy ∇2ψ=−λψ so it is an eigenfunction of the Laplacian. The vorticity equation then becomes ∂ω∂t=ν∇2ω=−λνω
Taylor–Green_vortex
Fluid dynamical instability
eddies. Important here is ψ, which is the absolute value of the complex eigenfunction of the stream function of the horizontal velocity. It represents the
Baroclinic instabilities in the ocean
Baroclinic_instabilities_in_the_ocean
Method of data analysis
functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al.,
Principal_component_analysis
Function in discrete mathematics
straightforward approach to obtain DFT eigenvectors is to discretize an eigenfunction of the continuous Fourier transform, of which the most famous is the
Discrete_Fourier_transform
Branch of mathematics
Fourier series is one instance of an eigenfunction expansion, with the exponentials einθ being the eigenfunctions of the rotation group acting on the circle
Mathematical_analysis
Different states of quantum systems
with a kinetic energy Hamiltonian operator using a wave function as an eigenfunction to obtain the energy levels as eigenvalues, but the Rydberg constant
Energy_level
Concept in radio communication
Fourier transform of a Dirac pulse is a complex exponential function, an eigenfunction of every linear system. The obtained channel transfer characteristic
Multipath_propagation
Area of mathematical analysis
of a function on Euclidean space into spherical harmonics, which are eigenfunctions of the Laplace operator. Abstract harmonic analysis extends the classical
Harmonic_analysis
Description of gauge theories using loop operators
Attempts have been made to describe gauge theories in terms of extended objects such as Wilson loops and holonomies. The loop representation is a quantum
Loop representation in gauge theories and quantum gravity
Loop_representation_in_gauge_theories_and_quantum_gravity
Raising and lowering operators in quantum mechanics
Levanthal, J. (2004). "Lenz vector operations on spherical hydrogen atom eigenfunctions". American Journal of Physics. 72 (8): 1013–1016. Bibcode:2004AmJPh
Ladder_operator
Quantum physics terminology
{\textstyle \Psi _{1}(x)=k\Psi _{2}(x)} which proves that the energy eigenfunction of a 1D bound state is unique. Furthermore it can be shown that these
Bound_state
Algebraic object with geometric applications
used to represent momentum fluxes Spherical tensor operators are the eigenfunctions of the quantum angular momentum operator in spherical coordinates Diffusion
Tensor
Polynomial sequence
{\displaystyle \operatorname {He} _{\lambda }(x)} may be understood as eigenfunctions of the differential operator L [ u ] {\displaystyle L[u]} . This eigenvalue
Hermite_polynomials
the Dirichlet problem on a bounded two-dimensional domain, the second eigenfunction has a nodal line that meets the boundary of the domain. The general
Nodal_line_conjecture
Mathematical theorem
b] consisting of eigenfunctions of TK such that the corresponding sequence of eigenvalues {λi}i is nonnegative. The eigenfunctions corresponding to non-zero
Mercer's_theorem
Special mathematical functions defined on the surface of a sphere
Maximum principle. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see Higher dimensions). A specific
Spherical_harmonics
Complex exponential in terms of sine and cosine
cosine. The reason for this is that the exponential function is the eigenfunction of the operation of differentiation. In electrical engineering, signal
Euler's_formula
in multipath channels. It is well known that sinusoidal signals are eigenfunctions of linear, and time-invariant systems. Therefore, if the channel is
Cyclic_prefix
mode. Mode shapes have a mathematical meaning as 'eigenvectors' or 'eigenfunctions' of the eigenvalue problem which arises, studying particular solutions
Modeshape
Technique for solving differential equations
for both differential operators, and T(t) and X(x) are corresponding eigenfunctions. We will now show that solutions for X(x) for values of λ ≤ 0 cannot
Separation_of_variables
Type of mathematical group
geometry states that there exists an orthonormal basis ϕ0,ϕ1,…,ϕn,… of eigenfunctions for Δ. The associated eigenvalues λ0=0<λ1⩽λ2⩽⋯ are unbounded and their
Arithmetic_Fuchsian_group
American mathematician
equations (International Press, 1995; 2nd ed., 2008) Hangzhou lectures on eigenfunctions of the Laplacian, (Annals of Mathematics Studies 188, Princeton University
Christopher_D._Sogge
Resistive and inductive circuit
is the angular frequency (in radians per second). The complex-valued eigenfunctions of any linear time-invariant (LTI) system are of the following forms:
RL_circuit
Method for solving certain nonlinear partial differential equations
operator M {\textstyle M} describes how the eigenfunctions evolve over time, and generates a new eigenfunction ψ ~ {\textstyle {\widetilde {\psi }}} of operator
Inverse_scattering_transform
State of matter at low temperatures
102.1189. T. D. Lee; K. Huang & C. N. Yang (1957). "Eigenvalues and Eigenfunctions of a Bose System of Hard Spheres and Its Low-Temperature Properties"
Superfluid_helium-4
Collection of mathematical theories
that to cover the phenomena one has already to deal with generalized eigenfunctions (for example, by means of a rigged Hilbert space). On the other hand
Spectral_theory
Property of a mass in motion
eigenvalue equation 𝐩ψ(p)=pψ(p), where the operator p acting on a wave eigenfunction ψ(p) yields that wave function multiplied by the eigenvalue p, in an
Momentum
Sound synthesis technique
into contributions from a set of modes (the spatial forms of which are eigenfunctions of the particular problem at hand, and are dependent on boundary conditions)
Additive_synthesis
Dimensionality reduction algorithm
demonstrated the DMD and related methods produce approximations of the Koopman eigenfunctions in addition to the more commonly used eigenvalues and modes. Residual
Dynamic_mode_decomposition
Foundational principle in quantum physics
general. Consider a particle in a one-dimensional box of length L. The eigenfunctions in position and momentum space are ψn(x,t)={Asin(knx)e−iωnt,0<x<L,0
Uncertainty_principle
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EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
EIGENFUNCTION
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