Localization of the Laplace operator eigenfunctions
By D. S. Grebenkov

The notion of localization was introduced in solid state physics 
by P. W. Anderson who suggested that randomness of a potential in 
the Schrodinger equation may lead to localized eigenstates in 
disordered media. This so-called Anderson localization, which is 
responsible for the metal/isolator transition in solids, has found 
numerous experimental confirmations. This phenomenon is called 
"strong" localization characterized by exponential space decay of 
the modulus of the wave amplitude. The localization has also been 
shown to occur for non-random potentials on self-similar 
deterministic fractals like Sierpinski gasket. In such disconnected 
domains, the localized eigenfunctions are supported by a small 
subset of the domain and are zero elsewhere.

A series of experimental and numerical studies have recently shown 
the emergence and importance of weakly localized eigenfunctions in 
irregularly-shaped domains. These eigenfunctions are localized in 
a small subset of the domain and are very small (but not zero) 
outside this subset. This type of localization, due to destructive 
interference is generally called "weak" localization and is 
characterized by a power law decay in space. It is worth stressing 
that the localized eigenfunctions constitute only a fraction among 
all eigenfunctions. The mathematical analysis of weakly localized 
eigenfunctions is therefore a challenging problem because it addresses 
only a subset of eigenfunctions. Even a rigorous mathematical definition 
of weak localization is not yet available, in spite of a growing 
interest to this topic in recent years.