eigenfunción

Eigenfunktion is a fundamental concept in mathematics and physics, particularly in the study of linear operators and differential equations. An eigenfunktion of a linear operator is a non-zero function that, when the operator is applied to it, results in a scalar multiple of the original function. This scalar multiple is known as the eigenvalue. Mathematically, if L is a linear operator and f is a function, then f is an eigenfunktion if L(f) = λf, where λ is the eigenvalue.

The concept of eigenfunctions is closely related to eigenvalues. Eigenfunctions represent the characteristic states or modes of a system that are left unchanged in direction by the operator, only scaled by the eigenvalue. In the context of differential equations, eigenfunctions often represent stable states or specific modes of vibration. For example, in quantum mechanics, the eigenfunctions of the Hamiltonian operator correspond to the stationary states of a quantum system, and their associated eigenvalues represent the possible energy levels of the system.

The set of all eigenfunctions of a given operator often forms a complete basis, meaning that any arbitrary function can be expressed as a linear combination of these eigenfunctions. This property makes eigenfunctions incredibly useful for solving complex problems by decomposing them into simpler, characteristic components. The process of finding eigenfunctions and eigenvalues is called spectral decomposition.