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Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space.

Peyerimhoff, N. and Täufer, M. and Veselić, I. (2017) 'Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space.', Nanosystems : physics, chemistry, mathematics., 8 (2). pp. 216-230.

Abstract

For the analysis of the Schrödinger 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 Schrödinger operators with random potentials. For discrete Schrödinger 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.

Item Type:Article
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.17586/2220-8054-2017-8-2-216-230
Date accepted:23 February 2017
Date deposited:01 November 2017
Date of first online publication:30 April 2017
Date first made open access:No date available

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