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.
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