Super-phenomena in arbitrary quantum observables

Superoscillations occur when a globally band-limited function locally oscillates faster than its highest Fourier coefficient. We generalize this effect to arbitrary quantum mechanical operators as a weak value, where the preselected state is a superposition of eigenstates of the operator with eigenvalues bounded to a range, and the postselection state is a local position. Superbehaviour of this operator occurs whenever the operator's weak value exceeds its eigenvalue bound. We give illustrative examples of this effect for total angular momentum and energy. In the later case, we demonstrate a sequence of harmonic oscillator potentials where a finite energy state converges everywhere on the real line, using only bounded superpositions of states whose asymptotic energy vanishes - "energy out of nothing". We resolve this paradoxical situation by showing the time-dependent Schr\"odinger equation no longer applies in the considered limit because of the non-commutation of the limit with the time-derivative. Nevertheless, this example demonstrates the possibility of mimicking a high-energy state with coherent superpositions of nearly zero-energy states for as wide a spatial region as desired.