The paper establishes new spectral inequalities for one-dimensional Schrödinger operators with growing potentials, with explicit exponents for thick and generalized thick sensor sets, based on a new quantitative Cauchy uniqueness estimate in the plane.
Spectral inequalities for Schr\"odinger equations with various potentials
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abstract
We study the spectral inequalities of Schr\"odinger operator in the whole space for different potentials, which can be power growth or continuously vanishing at infinity. The spectral inequalities quantitatively depend on the density of the sensor sets with positive measure, growth rate of the potentials and spectrum (or eigenvalues). One important component in the poof is the adaptation of propagation of smallness argument for gradients in \cite{LM18}. As an application, we apply the spectral inequalities to obtain quantitative observability inequalities for heat equations.
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Spectral inequalities for Schr\"odinger equations and quantitative propagation of smallness in the plane
The paper establishes new spectral inequalities for one-dimensional Schrödinger operators with growing potentials, with explicit exponents for thick and generalized thick sensor sets, based on a new quantitative Cauchy uniqueness estimate in the plane.