Hardy uniqueness principle for the linear Schrodinger equation on quantum regular trees

TL;DR

This paper investigates unique continuation properties of the linear Schrödinger equation on quantum regular trees, extending classical results to complex graph structures with infinite edges.

Contribution

It introduces new unique continuation results for the Schrödinger equation on regular trees and star-shaped graphs, bridging analysis on real lines and complex graph geometries.

Findings

Established unique continuation principles on regular trees.

Extended classical results to quantum graphs with infinite edges.

Analyzed Schrödinger equations with piece-wise constant coefficients and real potentials.

Abstract

In this paper we consider the linear Schrodinger equation (LSE) on a regular tree with the last generation of edges of infinite length and analyze some unique continuation properties. The first part of the paper deals with the LSE on the real line with a piece-wise constant coefficient and uses this result in the context of regular trees. The second part treats the case of a LSE with a real potential in the framework of a star-shaped graph.