Tuesday Seminar of Analysis

Seminar information archive ~05/18Next seminarFuture seminars 05/19~

Date, time & place Tuesday 16:00 - 17:30 156Room #156 (Graduate School of Math. Sci. Bldg.)
Organizer(s) ISHIGE Kazuhiro, SAKAI Hidetaka, ITO Kenichi

2013/12/10

16:30-18:00   Room #128 (Graduate School of Math. Sci. Bldg.)
Abel Klein (UC Irvine)
Quantitative unique continuation principle, local behavior of solutions, and bounds on the density of states for Schr\\"odinger operators (ENGLISH)
[ Abstract ]
We establish bounds on the density of states measure for Schr\\"odinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a ``density of states outer-measure'' that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-H\\"older continuity for this density of states outer-measure in one, two, and three dimensions for Schr\\"odinger operators, and in any dimension for discrete Schr\\"odinger operators. Our proofs use a quantitative unique continuation principle and the local behavior of approximate solutions of the stationary Schr\\"odinger equation.
(Joint work with Jean Bourgain.)
References: Jean Bourgain and Abel Klein: Bounds on the density of states for Schr\\"odinger operators. Invent. Math. 194, 41-72 (2013).