= Localized eigenstate
{title2=$H\psi=E\psi$}
A <localized eigenstate> is an <energy eigenstate> whose spatial weight remains concentrated near some position as the system grows. An exponentially <localized eigenstate> has a decaying envelope characterized by a <localization length>. A small <participation ratio> measures concentration in a chosen site basis; it does not by itself establish an exponential tail or a transport law.
An <energy eigenstate> changes only by an overall <wave phase> under time evolution, whether localized or extended. Its stationary probability distribution alone therefore does not distinguish transport mechanisms. The important distinction concerns spatial extent and how superpositions of <energy eigenstates> carry or retain a disturbance. A wave packet assembled from <localized eigenstates> with nearby centres can remain confined, whereas extended modes can support propagation.
Back to article page
