The area of low-dimensional quantum systems on discrete spaces is a rapidly growing research field lying at the interface between quantum theoretical developments, like discrete and q-difference equations, and tight binding superlattice models in solid-state physics. Systems on discrete spaces are promising candidates for applications in several areas. Indeed, the dynamic localization of electrons on the 1D lattice under the influence of an external electric field serves to describe time-dependent transport in quantum wires, linear optical absorption spectra, and the generation of higher harmonics. Odd-even parity effects and the flux dependent oscillations of total persistent currents in discretized rings can also be invoked. Technological developments are then provided by conductance calculations characterizing 1D conductors, junctions between rings and leads or rings and dots, and by quantum LC-circuits. Accordingly, the issues presented in this book are important starting points for the design of novel nanodevices.Contents:Lattice Structures and DiscretizationsPeriodic Quasiperiodic and Confinement PotentialsTime Discretization SchemesDiscrete Schrödinger Equations. Typical ExamplesDiscrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl AlgebrasHopping Hamiltonians. Electrons in Electric FieldTight Binding Descriptions in the Presence of the Magnetic FieldThe Harper-Equation and Electrons on the 1D RingThe q-Symmetrized Harper EquationQuantum Oscillations and Interference Effects in NanodevicesReadership: Physicists, graduate students in physics and solid state physics, researchers and engineers in the field of microelectronics.