Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators

TL;DR

This paper establishes an optimal bound on the vanishing order of solutions to certain subelliptic Schrödinger equations, providing a quantitative strong unique continuation property similar to classical Laplacian results.

Contribution

It introduces a frequency function approach for Baouendi-Grushin operators to derive quantitative bounds on solution vanishing orders, extending unique continuation results to subelliptic operators.

Findings

Optimal vanishing order bound for solutions

Quantitative strong unique continuation established

Extension of classical Laplacian results to subelliptic operators

Abstract

Based on a variant of the frequency function approach of Almgren, we establish an optimal bound on the vanishing order of solutions to stationary Schr\"odinger equations associated to a class of subelliptic equations with variable coefficients whose model is the so-called Baouendi-Grushin operator. Such bound provides a quantitative form of strong unique continuation that can be thought of as an analogue of the recent results of Bakri and Zhu for the standard Laplacian.