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We prove upper bounds on the number of resonances or eigenvalues ofSchr\"odinger operators $-\Delta+V$ with complex-valued potentials, where$d\geq 3$ is odd. The novel feature of our upper bounds is that they areeffective, in the sense that they only depend on an exponentially weightednorm. Our main focus is on $L^p$ potentials, but we also obtain new results forcompactly supported or pointwise decaying potentials. The main technicalinnovation, possibly of independent interest, are singular value estimates forFourier-extension type operators. The obtained upper bounds not only recoverseveral known results in a unified way, they also provide new bounds forpotentials which are not amenable to previous methods.

Effective upper bounds on the number of resonance in potential scattering