Disordered supersymmetric quantum mechanics

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

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=∫0m(dy). (ii) For singular Lévy measures, ∫0m(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).


Les commentaires sont fermés.