We addressed quantization phenomena in open systems and confined motion in low-dimensional systems, as well as quantized sources in 3-dimensions. The thesis of the paper is that if we simply cast the Bohr–Sommerfeld (B-S) quantization condition as a U(1) gauge theory, like the gauge field of Chern-Simons gauge theory or as in topological band theory (TBT) of condensed matter physics in terms of Berry connection and Berry curvature to make it self-consistent, then the quantization method in all the physical phenomena treated in this paper are unified in the sense of being traceable to the self-consistent B-S quantization. These are the stationary quantization of due to oscillatory dynamics in compactified space and time for steady-state systems (e.g., particle in a box or torus, Brillouin zone, and Matsubara time zone or Matsubara quantized frequencies), and the quantization of sources through the gauge field. Thus, the self-consistent B-S quantization condition permeates the quantization of integer quantum Hall effect (IQHE), fractional quantum Hall effect (FQHE), the Berezenskii-Kosterlitz-Thouless vortex quantization, Aharonov–Bohm effect, the Dirac magnetic monopole, the Haldane phase, contact resistance in closed mesoscopic circuits of quantum physics, and in the monodromy (holonomy) of completely integrable Hamiltonian systems of quantum geometry. In transport of open systems, we introduced a novel phase-space derivation of the quantized conductance of the IQHE based on nonequilibrium quantum transport and lattice Weyl transform approach.

We give a simple macroscopic phase-space explanation of fractional quantum Hall effect (FQHE), in a fashion reminiscent of the Landau-Ginsburg macroscopic symmetry breaking analyses. This is in contrast to the more complicated microscopic variational wavefunction approaches. Here, we employ a novel approach based on nonequilibrium quantum transport in the lattice Weyl transform formalism, coupled with the Maxwell Chern-Simons gauge theory for defining fractional filling of Landau levels. We show that flux attachment concept is inherent in fully occupied and as well as in partially occupied Landau levels. We derived the scaling k-factor in Chern-Simons gauge theory, as corresponding to the scaling of the magnetic fields or magnetic flux in FQHE. This is shown to be crucial in our simple explanation of FQHE as a topological invariant in phase space. The integer k this must be a prime number, whereas for fractions the numerator of k must also be a prime number. The assumption in the literature that the denominator of v = 1 k is given by the expression, (2n + 1), is wrong. Furthermore, although even denominators for v are speculated in the literature, this idea is definitely not allowed in our analysis.

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In this study, the Fourier series expansions of the Apostol-Frobenius type of Tangent and Genocchi polynomials of higher order are derived using the Cauchy residue theorem. Some novel and intriguing results are obtained by applying the Fourier series expansion of these types of polynomials.