Disorder, point scatterers and random matrix products

One-dimensional quantum Hamiltonians for random potentials made of generalised point scatterers

Spectral properties of a one-dimensional Schrödinger Hamiltonian for a potential given by a random superposition of delta potentials have been studied for a long time : Schmidt (1957) for delta scatterers at regularly spaced positions with random weights, Lax & Philips (1958), Frisch & Lloyd (1960) and Bychkov & Dykhne (1966) for random positions with fixed weights, and Nieuwenhuizen (1983) for random positions and random weights (See also the book by Lifshits, Gredeskul & Pastur, 1988).

However the delta potential is a particular realisation of « point scatterer »: a general point scatterer may be described by a 2*2 S-matrix. The unitarity of the S matrix implies that it can be parametrised by four real parameters (the group U(2)=U(1)*SU(2) has 4 generators). The case of delta scatterer corresponds to a one-parameter subgroup. We have considered models of random generalised point scatterers at random positions. The close relation with the study of products of random matrices is emphasized.

General products of random 2*2 matrices in the continuum limit (random matrices close to the identity)

This general formulation allows us to exhaust all possible 1D disordered models mapped onto random matrix products of SL(2,R) ; we have provided a classification of solutions that can be understand as a classification of 1D continuum disordered models. Alain Comtet, Jean-Marc Luck, Christophe Texier and Yves Tourigny, J. Stat. Phys. 150, 13-65 (2013) (and math-ph arXiv:1208.6430).

See also page “ products of random matrices

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