Effective upper bounds on the number of resonance in potential scattering

We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators $-\mathrm{\Delta }+V$ with complex-valued potentials, where $d\ge 3$ 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 $L^{(d+1)\mathrm{/}2,1\mathrm{/}2}$, 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.