A Note on a Capelli Operator and its Resonance

We study in parallel the resonances of the standard Laplacian on \(\mathbb {R}^3\) and of a Capelli operator on \(\mathbb {R}^2\) by exploiting their invariance under two appropriate group actions. In particular, we recover the well-known fact that the Laplacian on \(\mathbb {R}^3\) has no resonances and we conclude that the Capelli operator on \(\mathbb {R}^2\) has a single resonance at \(z=0\) . The full proof of this fact is contained in a joint paper with A. Pasquale and T. Przebinda (J Lie Theory 33(1):93–132, 2023).