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Its eigenfunctions form a basis of the function space on which the operator is defined [5] As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
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 multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
This function ω(λ) plays the role of the characteristic polynomial of D. Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of D and that each non-zero eigenspace is one-dimensional.
If the function κ is L 1 μ (X), where κ(x)=K(x,x), for all x in X, then there is an orthonormal set {e i} i of L 2 μ (X) consisting of eigenfunctions of T K such that corresponding sequence of eigenvalues {λ i} i is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on X and K has the representation
Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian.Ignoring complications about continuous spectra, we consider the discrete spectrum of and a basis of eigenvectors {| } (see spectral theorem for Hermitian operators for the mathematical background): | =, where is the Kronecker delta = {, =, and the {| } satisfy the eigenvalue equation | = | .
The covariance function K X satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λ k, e k (t) of eigenvalues and eigenfunctions of T K X forming an orthonormal basis of L 2 ([a,b]), and K X can be expressed as (,) = = ()
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue.
Let f be the characteristic function of the measurable set h −1 (λ), then by considering two cases, we find , () = (), so λ is an eigenvalue of T h. Any λ in the essential range of h that does not have a positive measure preimage is in the continuous spectrum of T h.