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.
- Alain Comtet, Christophe Texier and Yves Tourigny,
Products of random matrices and generalised quantum point scatterers
J. Stat. Phys. 140, 427-466 (2010)
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 ”