The Aharonov-Bohm effect on the sojourn time of the two-dimensional inverted harmonic oscillator under a perpendicular magnetic field

In the nonrelativistic or relativistic studies of the well-known effect of an infinitely long and infinitely thin solenoid containing magnetic flux φ, the so-called Aharonov-Bohm (AB) field, in terms of the Diracs fundamental unit of the magnetic flux φ₀, the magnetic flux φ is usually expressed as a flux quantum number, which can be decomposed into two parts: the integer and the mantissa part. It has been commonly accepted in the literature that the integer part is unimportant and only the mantissa determines all the physical effects. In this paper, by investigating the AB effect on the the sojourn time of a two-dimensional (2D) inverted harmonic oscillator under a perpendicular uniform magnetic field, we show that the sojourn time depends upon the magnitude, rather than solely the mantissa of the magnetic flux quantum number. This then provides an example that can manifest the quantum effect of the integer part of the flux quantum number, the reason behind is explained via gauge transformation.