Spectral inequalities for Schr\"odinger equations and quantitative propagation of smallness in the plane

This paper deals with spectral inequalities for one-dimensional Schr\"odinger operators with potentials bounded between two increasing functions (weights). The spectral inequality allows one to estimate the norm of a function with bounded spectrum by its values on a certain sensor set. We say that a measurable subset of the real line is thick if the measure of the intersection of this set with any interval of fixed length is bounded from below. First, we consider thick sensor sets a large class of pairs of weights. For potentials constrained between two polynomials, spectral inequalities for a broad class of so-called generalized thick sets are analyzed. A quantitative dependence of the constants in the spectral inequalities on the density of the sensor sets, the growth rate of the potentials, and the spectral interval is established. The proofs rely on a new quantitative propagation of smallness (or quantitative Cauchy uniqueness) for elliptic equations in the plane.