Effective upper bounds on the number of resonances in potential scattering

We prove upper bounds on the number of resonances and eigenvalues of Schrödinger operators −Δ + 𝑉 with complex-valued potentials, where 𝑑 ⩾ 3 is odd. The novel feature of our upper bounds is that they are 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 that are not amenable to previous methods.