On the vanishing of Green's function, desingularization and Carleman's method

TL;DR

This paper investigates the conditions under which the Green's function for the operator -Δ + V vanishes at singularities in three dimensions, providing bounds and enhancing understanding of unique continuation properties.

Contribution

It establishes an upper bound on the vanishing order of Green's functions with minimal assumptions on the potential V and improves results on strong unique continuation for eigenfunctions.

Findings

Derived an upper bound on Green's function vanishing order.

Improved results on strong unique continuation in 3D.

Provided minimal assumptions on potential V for analysis.

Abstract

The subject of the present paper is the phenomenon of vanishing of the Green function of the operator $-\Delta + V$ on $\mathbb R^3$ at the points where a potential $V$ has positive critical singularities. More precisely, imposing minimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upper bound on the order of vanishing of the Green function. As a by-product of our proof, we improve the existing results on the strong unique continuation for eigenfunctions of $-\Delta + V$ in dimension $d=3$.