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

Peyerimhoff, N.; Täufer, M.; Veselić, I.

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


Authors

M. Täufer

I. Veselić



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.

Citation

Peyerimhoff, N., Täufer, M., & Veselić, I. (2017). Unique continuation principles and their absence for Schrödinger eigenfunctions on combinatorial and quantum graphs and in continuum space. Nanosistemy: fizika, himiâ, matematika Наносистемы: физика, химия, математика (Print), 8(2), 216-230. https://doi.org/10.17586/2220-8054-2017-8-2-216-230

Journal Article Type Article
Acceptance Date Feb 23, 2017
Online Publication Date Apr 30, 2017
Publication Date Apr 30, 2017
Deposit Date Nov 1, 2017
Publicly Available Date Nov 1, 2017
Journal Nanosystems : physics, chemistry, mathematics
Print ISSN 2220-8054
Publisher St. Petersburg National Research University of Information Technologies, Mechanics and Optics
Peer Reviewed Peer Reviewed
Volume 8
Issue 2
Pages 216-230
DOI https://doi.org/10.17586/2220-8054-2017-8-2-216-230

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