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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).
The index j represents the jth eigenvalue or eigenvector and runs from 1 to . Assuming the equation is defined on the domain [,], the following are the eigenvalues and normalized eigenvectors. The eigenvalues are ordered in descending order.
As shown in an earlier example, the solution of Equation is the exponential = /. Equation is the time-independent Schrödinger equation. The eigenfunctions φ k of the Hamiltonian operator are stationary states of the quantum mechanical system, each with a corresponding energy E k. They represent allowable energy states of the system and may be ...
This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v. [37] [38]
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).
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 (,) = = () The process X t can be expanded in terms of the eigenfunctions e k as:
Due to the variational condition, Eq. (4), the second term in Eq. (5) vanishes. In one sentence, the Hellmann–Feynman theorem states that the derivative of the stationary values of a function(al) with respect to a parameter on which it may depend, can be computed from the explicit dependence only, disregarding the implicit one.
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 = +∞, + =