# Unique continuation principles and their absence for Schr\"odinger eigenfunctions on combinatorial and quantum graphs and in continuum space

## TL;DR

This paper investigates the presence or absence of unique continuation principles (UCP) for Schr"odinger eigenfunctions across various settings, highlighting their implications for spectral properties and the regularity of the integrated density of states.

## Contribution

It provides a comprehensive analysis of when UCP holds or fails for Schr"odinger operators on different graph structures and in continuum space, revealing new insights into spectral regularity.

## Key findings

- UCP holds in continuum space, leading to regular IDS.
- Weak UCP suffices for continuity of IDS on lattice models.
- UCP generally fails on combinatorial and quantum graphs, affecting spectral regularity.

## Abstract

For the analysis of the Schr\"odinger 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 Schr\"odinger 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.