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In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as = for some scalar eigenvalue . [1] [2] [3] The solutions to this equation may ...
The differential equation is said to be in Sturm–Liouville form or self-adjoint form.All second-order linear homogenous ordinary differential equations can be recast in the form on the left-hand side of by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector).
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.
The eigenvalue equation for D is the differential equation = () The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions . Derivative operator example
Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the spectral measure. The same formula had also been obtained independently by E. C. Titchmarsh in 1946 (scientific communication between Japan and the United Kingdom had been interrupted by World War II).
Let (H, , ) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator.Then there is a sequence of non-zero real eigenvalues λ i, i = 1, …, N, with N equal to the rank of A, such that |λ i | is monotonically non-increasing and, if N = +∞, + =
Let an eigenvalue equation be solved by linearly expanding the unknown function in terms of N known functions. Let the resulting eigenvalues be ordered from the smallest (lowest), λ 1, to the largest (highest), λ N. Let the same eigenvalue equation be solved using a basis set of dimension N + 1 that comprises the previous N functions plus an ...
This variational characterization of eigenvalues leads to the Rayleigh–Ritz method: choose an approximating as a linear combination of basis functions (for example trigonometric functions) and carry out a finite-dimensional minimization among such linear combinations. This method is often surprisingly accurate.
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