Spectral inequality for Schr\"odinger equations with power growth potentials

TL;DR

This paper establishes a spectral inequality for Schrödinger equations with power growth potentials, confirming a conjecture and linking the inequality to the decay of sensor sets and potential growth rates.

Contribution

It introduces a novel spectral inequality for Schrödinger equations with power growth potentials, utilizing advanced Carleman estimates that incorporate gradient information.

Findings

Spectral inequality depends on sensor set decay and potential growth rate.

Proof uses modified Carleman estimates with gradient information.

Confirms a conjecture from previous literature.

Abstract

We prove a spectral inequality for Schr\"odinger equations with power growth potentials, which particularly confirms a conjecture in \cite{DSV}. This spectral inequality depends on the decaying density of the sensor sets, and the growth rate of potentials. The proof relies on three-ball inequalities derived from modified versions of quantitative global and local Carleman estimates that take advantage of the gradient information of the potentials.