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

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

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