Lower bounds for resonance widths in potential and obstacle scattering
Consider the Schrödinger equation \((i\partial_ t+\Delta -V(x))\psi =0\), \(x\in {\mathbb{R}}^ 3\) and the wave equation \((\partial_ t^ 2- \Delta)u=0\) outside an obstacle \(\Omega\) in \({\mathbb{R}}^ 3\). It is assumed in both cases that the perturbation is supported in a ball \(B_{\rho}.\) The authors obtain an explicit lower bound for the size of the imaginary parts of the resonances \(k=\kappa -i\eta\) for this equation. The first main result deals with the Schrödinger operator at energies where classical trapping does not happen. It is shown that potentials supported in a ball \(B_{\rho}\), satisfying \(\kappa^ 2-V-r\partial V/\partial r>0\) do not produce resonances with \(\eta\) R very small and \(\kappa\) R not too small. For \(\kappa\) large condition becomes approximately \(4.72\eta R<1.\) The second result deals with potentials for which the non-trapping condition fails. Then resonances with \(\eta R\ll 1\) are expected. The authors give lower bounds which are exponentially small in a certain quantity which roughly measures the size of the trapping barrier. Finally the authors give a lower bound of \(\eta\) for the Laplacian in the exterior of a star-shaped obstacle. This lower bound approaches \(R^{- 1}(2+e)^{-1}\) as \(\kappa\to \infty\).
scientific article; zbMATH DE number 4171520