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...

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}}}...

Heath-Brown (1986) Eigenfunction Expansions Associated with Second-order Differential Equations. Part I (1946) 2nd. edition (1962); Eigenfunction Expansions Associated...

energies). It is also called energy eigenvector, energy eigenstate, energy eigenfunction, or energy eigenket. It is very similar to the concept of atomic orbital...

mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint...

to multiplication in the frequency domain. For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials...

complete orthonormal system of eigenfunctions for the Fourier transform on L2(R). However, this choice of eigenfunctions is not unique. Because of F 4...

y=y(x)} of the problem. Such functions y {\displaystyle y} are called the eigenfunctions associated to each λ {\displaystyle \lambda } . Sturm–Liouville theory...

{\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 =...

In applied mathematics, the Hough functions are the eigenfunctions of Laplace's tidal equations which govern fluid motion on a rotating sphere. As such...

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...

is an orthonormal basis of the Hilbert space L2 that consists of the eigenfunctions of the autocovariance operator. FPCA represents functional data in the...

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...

function gives an eigenfunction with eigenvalue 1/2 and multiplicity one; that there are no corresponding generalized eigenfunctions with eigenvalue 1/2;...

eigenfunction solutions / eigenvector solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions /...

potential system, if a wavefunction ψ ( r ) {\displaystyle \psi (r)} is an eigenfunction of the Hamiltonian operator H ^ ( p ^ , x ^ ) {\displaystyle {\hat {H}}({\hat...

Hamiltonian. For a spin half nucleus there are two eigenfunctions for ÎZ. Let m = +1/2 and m = -1/2 and eigenfunctions are, ÎZ ψm = mħψm Applying the equation of...

problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary...

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...

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...

Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes...

estimate (from above) for Hausdorff measures on the zero sets of Laplace eigenfunctions defined on compact smooth manifolds and an estimate (from below) in...

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...

operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with: − Δ f = λ f . {\displaystyle -\Delta f=\lambda f.} This is known...

quantum problems involving rotational symmetry. Being an observable, its eigenfunctions represent the distinguishable physical states of a system's angular...

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...

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...

ψ λ ⟩ {\displaystyle |\psi _{\lambda }\rangle } , is an eigenstate (eigenfunction) of the Hamiltonian, depending implicitly upon λ {\displaystyle \lambda...

(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 ,...

spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In...