The wave functions, the autocorrelation functions of which decay faster than
t−2, for both the one-dimensional free particle system and the repulsive-potential system are examined. It is then shown that such wave functions constitute a dense subset of
L2(R1), under several conditions that are particularly satisfied by the square barrier potential system. It implies that the faster than
t−2-decay character of the autocorrelation function persists against the perturbation of potential. It is also seen that the denseness of the above subset is guaranteed by that of the domain of the Aharonov-Bohm time operator.
We explore a mesoscopic system made of a long nonconducting cylinder of inner radius [Formula: see text] where a spinless electron interacts with an attractive [Formula: see text] potential. We analyze the asymptotic behavior of the radial wave function by assuming that there is an impenetrable potential wall located at [Formula: see text], and thus, discuss the effects of rotation on the energy levels, as well as the Aharonov–Bohm effect for bound states. Further, we include a non-Hermitian operator and show that the non-Hermitian Hamiltonian has real energy eigenvalues. We also discuss the effects of rotation and the Aharonov–Bohm effect from the real energy eigenvalues. We extend our discussion to the revival time.
Recent neutron interferometry experiments have been interpreted as demonstrating a new topological phenomenon similar in principle to the usual Aharonov-Bohm (AB) effect, but with the neutron's magnetic moment replacing the electron's charge. We show that he new phenomenon, called scalar AB (SAB) effect, follows from an ordinary local interaction, contrary to the usual AB effect, and we argue that the SAB effect is not a topological effect by any useful definition. We find that SAB actually measures an apparently novel spin autocorrelation whose operator equations of motion contain the local torque in the magnetic field. We note that the same remarks apply to the Aharonov-Casher effect.