A remark about internal Lifshitz asymptotes (Q1608687)

Let \(H_\omega =H+V_\omega\) be a random operator acting on \(L^2(\mathbb R^d)\) where \(H\) is a \(\mathbb Z^d\)-periodic Schrödinger operator, \(V_\omega (x)=\sum_{\gamma \in \mathbb Z^d}\omega_\gamma V(x- \gamma)\), \(\omega_\gamma\) are i.i.d. bounded nonnegative random variables, \(V(x)\geq 0\) has a power-like decay at infinity. The author studies the asymptotic behavior of the integrated density of states \(N_p(E)\) of the operator \(H_\omega\), as \(E\to +0\). The crucial quantity is the number \[ \kappa =-\lim\limits_{E\to +0}\frac{\log |\log \mathbb P\{ \omega_0\leq E\}|}{\log E}. \] Depending on the value of \(\kappa\), the asymptotics of \(N_p(E)\) is different, being interpreted as a representation of the classical or quantum regime.