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Regularity of the Density of States of Random Schrödinger Operators
D.R. Dolai, , A. Mallick
Published in Springer
Volume: 378
Issue: 1
Pages: 299 - 328
In this paper we solve a long standing open problem for Random Schrödinger operators on L2(Rd) with i.i.d single site random potentials. We allow a large class of free operators, including magnetic potential, however our method of proof works only for the case when the random potentials satisfy a complete covering condition. We require that the supports of the random potentials cover Rd and the bump functions that appear in the random potentials form a partition of unity. For such models, we show that the Density of States (DOS) is m times differentiable in the part of the spectrum where exponential localization is valid, if the single site distribution has compact support and has Hölder continuous m+ 1 st derivative. The required Hölder continuity depends on the fractional moment bounds satisfied by appropriate operator kernels. Our proof of the Random Schrödinger operator case is an extensions of our proof for Anderson type models on ℓ2(G) , G a countable set, with the property that the cardinality of the set of points at distance N from any fixed point grows at some rate in Nα, α> 0. This condition rules out the Bethe lattice, where our method of proof works but the degree of smoothness also depends on the localization length, a result we do not present here. Even for these models the random potentials need to satisfy a complete covering condition. The Anderson model on the lattice for which regularity results were known earlier also satisfies the complete covering condition. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature.
About the journal
JournalData powered by TypesetCommunications in Mathematical Physics
PublisherData powered by TypesetSpringer