Cf. the review on SUSY QM : page on review articles
How to break supersymmetry with a random scalar potential
Cf. paragraph on supersymmetric quantum mechanics
- Christian Hagendorf and Christophe Texier,
Breaking of supersymmetry in one-dimensional a random Hamiltonian
J. Phys. A: Math. Theor. 41, 405302 (2008). (32pp)
cond-mat arXiv:0805.2883. - Christophe Texier and Christian Hagendorf,
One-dimensional classical diffusion in a random force field with weakly concentrated absorbers
Europhys. Lett. 86, 37011 (2009). (6pp)
cond-mat arXiv:0902.2698. - Aurélien Grabsch, Christophe Texier and Yves Tourigny,
One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers
J. Stat. Phys. 155, 237-276 (2014)
cond-mat arXiv:1310.6519.
See also the page on Classical diffusion in a random force field (Sinai problem)
One-dimensional supersymmetric Hamiltonian with Lévy noises
In this work we consider Schrödinger supersymmetric Hamiltonians when the function φ(x) is a Lévy noise, i.e. when Φ(x)=∫0x dx’ φ(x’) is a Lévy process (a random process generalising the Brownian motion). The case of subordinators is considered (non decreasing processes). We have discovered a new exact model for a Lévy process with singular Lévy measure m(dy). Moreover, we have provided a general discussion of low-energy spectral properties for arbitrary subordinators. (i) For regular Lévy measure we show that the main exponential behaviour of the integrated density of states is N(E) ∼ exp[-πρ/√E] where ρ=∫0∞m(dy). (ii) For singular Lévy measures, ∫0∞m(dy)=∞, we obtain N(E) ∼ exp[-C E-η/2] where the exponent is related to the singularity of the Lévy measure by η=1/(1-α), where m(dy) ∝ y-1-αdy when y → 0+ (for 0<α<1).
- Alain Comtet, Christophe Texier and Yves Tourigny,
Supersymmetric quantum mechanics with Lévy disorder in one-dimension
J. Stat. Phys. 145, 1291-1323 (2011)
math-ph arXiv:1105.5506