najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2017-8-2-216-230najo-653Research ArticleMATHEMATICSМАТЕМАТИКАUnique continuation principles and their absence for Schrodinger eigenfunctions on combinatorial and quantum graphs and in continuum spacePeyerimhoffN.

Department of Mathematical Sciences

norbert.peyerimhoff@durham.ac.ukTauferM.

Fakultat fur Mathematik

mtaeufer@math.tu-dortmund.deVeselicI.

Fakultat fur Mathematik

iveselic@math.tu-dortmund.deDurham UniversityUnited KingdomTechnische Universitat DortmundGermany20171208202582216230Copyright © Peyerimhoff N., Taufer M., Veselic I., 20252025Peyerimhoff N., Taufer M., Veselic I.Peyerimhoff N., Taufer M., Veselic I.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/653

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.

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