On the Continuity of the Integrated Density of States in the Disorder

Abstract

Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators, and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on R d .

Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models

“…Another implication of Theorem 4.1 for random Schrödinger operators on the lattice is Lipschitz continuity of the DOSm in the underlying single-site probability measure. This is an improvement of [11,Theorem 3.1], where Hölder continuity was proven, and recovers the results of Shamis [16] and of Kachkovskiy [13].…”

“…Whereas our earlier work focused on random Schrödinger operators, we extend these results to Schrödinger operators on infinite graphs with deterministic potentials and ergodic potentials, and improve our results for random potentials. In particular, we prove the Lipschitz continuity of the DOSm for random Schrödinger operators on the lattice, recovering results of [13,16]. For our treatment of deterministic potentials, we first study the density of states outer measure (DOSoM), defined for all Schrödinger operators, and prove a deterministic result of the modulus of continuity of the DOSoM with respect to the potential.…”

On the abominable properties of the almost Mathieu operator with well approximated frequencies

“…Our next proposition, possibly of independent interest, is a counterpart of this fact for the integrated density of states; it quantifies a result of Avron and Simon [65,Theorem 3.3]. We mention similar-looking results for random Schrödinger operators recently obtained in [43,66].…”

Dependence of the Density of States on the Probability Distribution. Part II: Schrödinger Operators on $$\pmb {\mathbb {R}}^d$$ and Non-compactly Supported Probability Measures

“…Since our first paper [15] was posted, we received comments from I. Kachkovskiy [16] in which he indicated a different proof for models on Z d with single-site probability measures of compact support that gives Lipschitz continuity of the DOSm. Independently, M. Shamis communicated another proof for the models considered here using different methods [19] also giving Lipschitz continuity for the DOSm with respect to the single-site probability measure. Shamis uses the Kantorovich-Rubinstein metric instead of the bounded Lipschitz metric (1.14).…”