Unique continuation principles and their absence for Schrodinger eigenfunctions on combinatorial and quantum graphs and in continuum space
https://doi.org/10.17586/2220-8054-2017-8-2-216-230
Abstract
For the analysis of the Schrodinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrodinger operators with random potentials. For discrete Schr odinger equations on the lattice, only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs, this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.
Keywords
eigenfunctions,
unique continuation,
Schrodinger equation,
Wegner estimate,
Integrated density of states.
Supporting agencies: This work was partially financially supported by the Deutsche Forschungsgemeinschaft through the grants VE 253/6-1 Unique continuation principles and equidistribution properties of eigenfunctions and VE 253/7-1 Multiscale version of the Logvinenko-Sereda Theorem. While writing part of this article, NP and MT enjoyed the hospitality of the Isaac Newton Institute during the programme Non-Positive Curvature Group Actions and Cohomology, supported by the EPSRC Grant EP/K032208/1. We would like to thank Michela Egidi for reading a previous version of the manuscript
About the Authors
N. Peyerimhoff
Durham University
United Kingdom
United Kingdom
Department of Mathematical Sciences
M. Taufer
Technische Universitat Dortmund
Germany
Germany
Fakultat fur Mathematik
I. Veselic
Technische Universitat Dortmund
Germany
Germany
Fakultat fur Mathematik
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For citations:
Peyerimhoff N., Taufer M., Veselic I. Unique continuation principles and their absence for Schrodinger eigenfunctions on combinatorial and quantum graphs and in continuum space. Nanosystems: Physics, Chemistry, Mathematics. 2017;8(2):216-230. https://doi.org/10.17586/2220-8054-2017-8-2-216-230
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