Question: If the wave function $\psi \to 0$ (pointwise or uniformly in mean), then does $|\psi|^2$ converge in quadratic mean (i.e. in $L^2$) to $0$?
Is this the source of the term "wave function"?
(Since the energy of a wave is proportional to the square of its amplitude, so if the expected energy of a wave goes to zero, then its ensemble amplitude goes to zero in quadratic mean. Likewise, if the expectation of a wave function goes to zero, then its probability density will decay to zero in $L^2$/quadratic mean. And the wave function is supposed to have something to do with energy, hence why it being square-integrable has something to do with it having finite energy, and thus in a certain sense is related to an elementary wave.)
Note: This is a follow-up to my previous question: Do waves and oscillations dissipating energy decrease in amplitude in mean square?