Effective upper bounds on the number of resonance in potential scattering

TL;DR

This paper establishes effective upper bounds on the number of resonances and eigenvalues for Schrödinger operators with complex potentials in odd dimensions, depending only on an exponentially weighted norm of the potential.

Contribution

It introduces new effective upper bounds that depend solely on an exponentially weighted norm of the potential, applicable to various classes of potentials.

Findings

Bounds depend only on exponentially weighted norm of V

Results apply to Lorentz space potentials and others

Unifies and extends previous resonance bounds

Abstract

We prove upper bounds on the number of resonances and eigenvalues of Schr\"odinger operators $-\Delta+V$ with complex-valued potentials, where $d\geq 3$ is odd. The novel feature of our upper bounds is that they are \emph{effective}, in the sense that they only depend on an exponentially weighted norm of V. Our main focus is on potentials in the Lorentz space $L^{(d+1)/2,1/2}$, but we also obtain new results for compactly supported or pointwise decaying potentials. The main technical innovation, possibly of independent interest, are singular value estimates for Fourier-extension type operators. The obtained upper bounds not only recover several known results in a unified way, they also provide new bounds for potentials which are not amenable to previous methods.