Clean function

In Mathematical, a clean function F of a linear Opérateur has on a Espace of functions is a clean Vecteur of the linear operator. It is a function not identically null and satisfactory:

$\ mathcal has F = \ lambda f$

for a Scalar λ , the Eigenvalue associated with F . The existence of clean vectors is typically of great help to analyze has .

For example, for any reality $\ \ alpha$ , $f_ \ alpha: \ R \ to \ R, \, X \ mapsto e^ {\ alpha X}$ is a clean function for the differential Opérateur

$\ mathcal has = \ frac {d^2} {dx^2} - \ frac {D} {dx},$

with like corresponding eigenvalue $\ \ lambda = \ alpha^2 - \ alpha$ . The clean functions play a big role in quantum Mécanique, where the equation of Schrödinger:

$i \ hbar \ frac {\ partial} {\ partial T} \ psi = \ mathcal H \ psi$

has solutions of the form:

$\ psi \ left (T \ right) = \ sum_k e^ {- I E_k T \ hbar} \ phi_k,$

where the $\ \ phi_k$ are clean functions of the operator $\ mathcal H$ with the eigenvalues $\ E_k$ . Because of the nature of the Hamiltonian operator $\ mathcal H$ , its own functions are Orthogonal are. That is not necessarily the case for the clean functions of other operators (like the example $\ A$ mentioned Ci-high).

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