By Erhardt Papp, Codrutza Micu

The realm of low-dimensional quantum structures on discrete areas is a swiftly transforming into study box mendacity on the interface among quantum theoretical advancements, like discrete and q-difference equations, and tight binding superlattice versions in solid-state physics. platforms on discrete areas are promising applicants for purposes in numerous parts. certainly, the dynamic localization of electrons at the 1D lattice lower than the effect of an exterior electrical box serves to explain time-dependent delivery in quantum wires, linear optical absorption spectra, and the new release of upper harmonics. Odd-even parity results and the flux based oscillations of overall continual currents in discretized jewelry is usually invoked. Technological advancements are then supplied by way of conductance calculations characterizing 1D conductors, junctions among jewelry and leads or earrings and dots, and via quantum LC-circuits. hence, the problems awarded during this ebook are very important beginning issues for the layout of novel nanodevices

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52) in which τ plays the role of an “elementary” time scale. g. Pokrowski (1928) and Beck (1929)) and especially on the ”chronon” hypothesis” forwarded by Caldirola (1976). Moreover, accounting for discrete time effects results in certain corrections to the relativistic mass energy formula (Wolf (1989)). Of course the “elementary” time can also be approached via τ = a/c, where a plays the role of an “elementary” length. 55) as usual. 51) becomes where ωnm = 1 (En − Em ) . 57) where exp (−knm τ ) = −iτ ωnm ± 2 τ2 1 − ωnm .

One sees that there is a π phase-difference in the oscillations characterizing these currents. 116) February 8, 2007 9:47 38 World Scientific Book - 9in x 6in papp˙micu˙world Low-Dimensional Nanoscale Systems on Discrete Spaces Fig. 2 The flux dependence of Cg (β, Ne ) for Ne = 3. 32. =− → e e2 − 2 ∇ arg ψ + A |ψ| 2m0 m0 c → − where A denotes the vector potential and ψ = |ψ| arg ψ. Details concerning this rather special device have been presented before (Ziman (1964)). 117) we can account both for the influence of a transversal and homogeneous magnetic field B as well as for the axial AB -flux ΦA .

Indeed, in order to reproduce in practice irrational frequencies an infinite experimental resolution would be necessary, but this latter point looks rather questionable. On the other hand there are fractal-like experimental realizations which can be viewed as competition effects between rational and irrational frequencies, in which case successive resolutions with increasing accuracy are in order. It is also clear that potentials on a discrete space for which the period is incommensurate with the spacing of the underlying lattice are able to produce nontrivial manifestations.

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