Wave decay outside two-dimensional compactly supported scatterers holds logarithmically without requiring spectral regularity at zero, with explicit zero-energy contributions.
Persistence and disappearance of negative eigenvalues in dimension two
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abstract
We compute asymptotics of eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schr\"odinger operators in dimension two. We distinguish persistent eigenvalues, which have associated resonances, from disappearing ones, which do not. We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit--Wigner peaks. We prove all of our results for circular wells, and extend some of them to more general problems using recent resolvent techniques.
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Unconditional wave decay in dimension two
Wave decay outside two-dimensional compactly supported scatterers holds logarithmically without requiring spectral regularity at zero, with explicit zero-energy contributions.