Regularity of the Surface Density of States

Abstract

dedicated to jean michel combes on the occasion of his 60 th birthdayWe prove that the integrated surface density of states of continuous or discrete Anderson-type random Schrödinger operators is a measurable locally integrable function rather than a signed measure or a distribution. This generalizes our recent results on the existence of the integrated surface density of states in the continuous case and those of A. Chahrour in the discrete case. The proof uses the new L p -bound on the spectral shift function recently obtained by Combes, Hislop, and Nakamura. Also we provide a simple proof of their result on the Hölder continuity of the integrated density of bulk states.

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

“…A number of further results on the spectral averaging and its applications to random Schrödinger operators can be found in [9], [12], [13], [19], [21], [33,Section 3], [39], [40], [43].…”

“…An application of the Birman-Solomyak formula to spectral averaging can be found in Section 3 of [33].…”

Statistical ensembles and density of states

“…For easier, recent proofs that they are Hölder continuous of any index smaller than 1, see [13,48]. Unfortunately, in general the Riemann-Stieltjes convolution does not improve Hölder continuity or any other similar kind of regularity (see, e.g., [24,17]).…”

“…This theory has been investigated intensively in the last years by mathematical physicists in the context of one-particle Schrödinger operators, in particular, operators with a periodic potential and random Schrödinger operators like Anderson type models. For recent references consult, e.g., [46,13,48,33] and for references before 1992 [27,41,10,61].…”

“…For Anderson-type random Schrödinger operators the existence of the density of states (i.e., the absolute continuity of the integrated density of states) is a complicated, still not completely solved problem. Very little known is known about regularity properties of the density of states [12,13,48,33].…”