It is a well-known result of W. Craig and B. Simon that the integrated density of states (IDS), \(k(\cdot)\), associated to an ergodic Jacobi matrix is log-Hölder continuous, that is, \[ |k(E)-k(E')|\leq CF(|E-E'|), \quad |E-E'|\leq \frac{1}{2}, \] for a suitable \((E,E')\)-independent constant \(C>0\) and \(F(\varepsilon)=[\log(\varepsilon^{-1})]^{-1}\), \(0<\varepsilon\leq 1/2\) [Duke Math. J. 50, 551–560 (1983; Zbl 0518.35027)].
In this paper, by considering operators of the form \(\Delta+V\) on \(\ell^2(\mathbb{Z})\), where \(\Delta\) denotes the discrete Laplacian and \(V\in \ell^{\infty}(\mathbb{Z})\) is limit-periodic (i.e., \(V\) is the limit in \(\ell^{\infty}\) norm of periodic potentials), the authors show that log-Hölder continuity of the IDS is optimal in the sense that the function \(F\) above cannot be replaced by another function which goes to zero faster as \(\varepsilon\downarrow 0\). To be more precise, let \(k_{V}(\cdot)\) denote the IDS of \(\Delta+V\) with \(V\) limit-periodic, that is, \[ k_V(E)=\lim_{N\rightarrow \infty}\frac{1}{N}\text{tr}\big(P_{(-\infty,E)}(H_{[0,N-1]}) \big), \] with \(H_{[0,N-1]}\) denoting the restriction of \(\Delta+V\) to \(\ell^2([0,N-1])\) and \(P_{(-\infty,E)}(H_{[0,N-1]})\) the spectral projection associated to \((-\infty,E)\); the authors prove:
Given any increasing function \(\phi:\mathbb{R}^+\rightarrow \mathbb{R}^+\) satisfying \(\lim_{x\rightarrow 0}\phi(x)=0\) and a constant \(C_0>0\), there is a limit-periodic potential \(V\in \ell^{\infty}(\mathbb{Z})\) satisfying \(\|V\|_{\infty}\leq C_0\) such that \[ \limsup_{E\rightarrow E'}\frac{|k_V(E)-k_V(E')|\log(|E-E'|^{-1})}{\phi(|E-E'|)}=\infty, \quad E\in \sigma(\Delta+V). \]
Consequently, one cannot have \[ |k_V(E)-k_V(E')|\leq C\cdot \frac{\phi(|E-E'|)}{\log(|E-E'|^{-1})} \] for any \(C\) and all \(V\).