Can someone please explain the following in mathematical language? What are angular excitations and s-waves mathematically?
"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
Edit: change in the title.
When you solve the central force Schrodinger equation, you can use separation of variables, that is, you assume that the solution is a product of a term that is purely radial ( = dependent on the distance $r$ from the central force origin) and purely angular , i.e. dependent purely on the angles $\phi, \theta$. This gives eigen-solutions that have the form $\Psi=A(r)B(\phi, \theta)$. Once this is done, it is natural to consider solutions where the angular term is a constant, that is $\Psi = A(r)$. These are the S waves. "Angular excitations" are how the angular terms affect the solution eigenvalues.