Question

Random quantum systems with asymmetric Lifshitz tails?

Answer 1

The asymmetric diffusion of a particle in a random one-dimensional medium can be described by a model of random potential with positive spectrum closely linked to supersymmetric quantum mechanics. We obtain analytical expressions for the density of states ρ (ε)(inverse relaxation time spectrum). This allows us to compute the averaged probability of return at any time. At zero energy ρ (ε) exhibits a variety of singular behaviours with a continuously varying exponent. This corresponds to the different phases of the diffusion problem at large time, including Sinaï's behaviour lang x 2 (t) rang= C ln 4 t. The validity of the dynamical-scaling assumption is discussed.

Answer 2

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) , let be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum that are less than or equal to . (Or see the rigorous definition in Sec. 2 of Ref. 1.)
The Lifshitz exponent is defined near the infimum as (e.g. in Ref. 1):
and similarly near the supremum
, the analogous exponent can be defined as
These exponents measure how fast the density of eigenvalues grows or decays near the extreme ends of the spectrum.
I have some empirical data for a disordered quantum mechanical system for which I appear to measure . This behavior does not seem to be observed in the model Hamiltonians that have been summarized in the review in Ref. 1 or in its references. I haven't found any literature that describes such behavior in any theoretical studies of quantum mechanical systems or random Schrödinger operators.
As I am not an expert in this field, I would appreciate any pointers to references where such asymmetric behavior in the upper and lower tails of have been observed.