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Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves...", c.f. first answer in https://physics.stackexchange.com/questions/370901/solution-for-inverse-square-potential-in-d-3-spatial-dimensions-in-quantum-mec

Is that assertion also valid in $N$ dimensions? And more generally for a Schrodinger operator in a rotationally symmetric riemannian manifold with radial potential? In that case, how to prove it? (references are welcome).

In fact, I would like to apply that kind of results in the study of stability of minimal surfaces. More details and context in this related question: Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions

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    $\begingroup$ Yes. If you have a rotationally symmetric Riemannian manifold with a radial potential, so you have a Schrodinger operator $H = -\Delta + V$, then $\langle f, Hf\rangle \geq \langle \bar{f}, H\bar{f}\rangle$ where $\bar{f}$ is the spherical average of $f$. $\endgroup$
    – Willie Wong
    Commented Jul 29, 2020 at 1:26

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