= Generalized eigenfunction
{title2=$Lu=\lambda u$}
A generalized eigenfunction solves the <eigenvalue equation> for a <linear operator> in an enlarged space, often a space of <distributions>, although it may fail to belong to the underlying <Hilbert space>. A <plane wave> $e^{ikx}$ on the whole <real line> is a generalized <momentum eigenstate>, with <momentum> $\hbar k$, and is not <square-integrable>. This usage differs from a <generalized eigenvector> in a <Jordan chain>, where a higher power of $L-\lambda$ vanishes rather than $L-\lambda$ itself.
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