Some properties of the one-dimensional Schrödinger operator with random potential — Anderson localisation
Introduction : Real systems generally possess a certain amount of disorder (impurities or structural defects). Whether the disorder strongly affects the physical properties is sometimes a difficult question. It is well known that wave phenomenon are particularly sensitive to the presence of disorder. A striking manifestation of disorder is the localization of the electronic waves: whereas in a perfect cristal the eigenstates are extended Bloch waves, a sufficiently strong disorder can localize the electronic wave functions, as shown by Anderson in a pioneering article in 1958.
Dimensionality plays a crucial role. Strictly one-dimensional systems are somehow less rich than two- and three-dimensional cases: the rigorous proof of localization of all eigenstates in 1d, by Goldsheit, Molchanov & Pastur (1977) forbids the existence of a localization-delocalization transition in the spectrum, as in 3d (Note however that delocalization can occur in 1d if the random potential possesses some particular correlations: such an example is provided by the disordered SUSY quantum mechanics). The fact that the elastic mean free path is of the same order as the localization length in 1d leaves not room for the existence of a diffusive regime(1). However, despite they are less rich, strictly 1d disorder systems offer the possibility to use much more powerful (nonperturbative) techniques and study much finer properties. Several examples of such studies are developed in the following articles.
(1) The disorder can be characterized by the elastic mean free path le which gives the length over which the momentum has relaxed. Another important length is the localization length lloc that measures the exponential damping of the wave functions. If we call L the system size we can distinguish several regimes for a weak disorder:
- The ballistic regime, when L << le, lloc
- The diffusive regime, when le << L << lloc (weak localization).
- The localized regime, when le, lloc << L(strong localization).
My research has been developed in the following directions: