Lifschitz tail in a magnetic field: coexistence of classical and quantum behavior in the borderline case

TL;DR

This paper precisely characterizes the low-energy behavior of the density of states in a magnetic field with impurities, confirming the coexistence of classical and quantum effects in the borderline case.

Contribution

It provides the first exact asymptotic analysis for the Lifschitz tail in the borderline case, completing the understanding of classical-quantum coexistence.

Findings

Confirmed the coexistence of classical and quantum regimes in the borderline case.

Derived exact low-energy asymptotics of the integrated density of states.

Resolved the last open case in the theory of magnetic Lifschitz tails.

Abstract

We establish the exact low-energy asymptotics of the integrated density of states (Lifschitz tail) in a homogeneous magnetic field and Poissonian impurities with a repulsive single-site potential of Gaussian decay. It has been known that the Gaussian potential tail discriminates between the so-called "classical" and "quantum" regimes, and precise asymptotics are known in these cases. For the borderline case, the coexistence of the classical and quantum regimes was conjectured. Here we settle this last remaining open case to complete the full picture of the magnetic Lifschitz tails.