The borderline Lifshits-tail asymptotic for algebraically decaying single-impurity potentials

Let $U$ be a non-negative single-impurity potential in dimension $d$, let $N(E)$ denote the integrated density of states, let $\rho$ be the impurity density, and let $\kappa_d$ and $C_d(d+2,\rho)$ be the constants appearing in the Lifshits-tail asymptotics. Suppose that

$$\lim_{|x|\to\infty}|x|^{d+2}U(x)=g$$

for some constant $g>0$.

Borderline Lifshits-tail conjecture. As $E\downarrow0$,

$$\log N(E)\sim-\left[\rho^{\frac{2}{d+2}}\left(\frac{\kappa_d}{2E}\right)^{\frac{d}{d+2}}+\bigl[C_d(d+2,\rho)\bigr]^{\frac{2}{d+2}}\left(\frac{g}{E}\right)^{\frac{d}{d+2}}\right]^{\frac{d+2}{2}}.$$

The case of algebraic decay with exponent $\alpha=d+2$ is identified as open because it lies at the transition between the quantum and classical Lifshits-tail regimes. The proposed formula describes the competition between the quantum kinetic contribution and the contribution from the long-range decay of the single-impurity potential.