Contents -- Preface -- 1. Lattice Structures and Discretizations -- 1.1 Discrete derivatives -- 1.2 The Jackson derivative -- 1.3 The q-integral -- 1.4 Generalized q-hypergeometric functions -- 1.5 The discrete space-time: a short retrospect -- 1.6 Quick inspection of q-deformed Schrodinger equations -- 1.7 Orthogonal polynomials of hypergeometric type on the discrete space -- 2. Periodic Quasiperiodic and Confinement Potentials -- 2.1 Short derivation of the Bloch-theorem -- 2.2 The derivation of energy-band structures -- 2.3 Direct and reciprocal lattices -- 2.4 Quasiperiodic potentials -- 2.5 A shorthand presentation of the elliptic Lame-equation -- 2.6 Quantum dot potentials -- 2.7 Quantum ring potentials -- 2.8 Persistent currents and magnetizations -- 2.9 The derivation of the total persistent current for electrons on the 1D ring at T =0 -- 2.10 Circular currents -- 3. Time Discretization Schemes -- 3.1 Discretized time evolutions of coordinate and momentum observables -- 3.2 Time independent Hamiltonians of hyperbolic type -- 3.3 Time independent Hamiltonians of elliptic type -- 3.4 The derivation of matrix elements -- 3.5 Finite difference Liouville-von Neumann equations and "elementary" time scales -- 3.6 The q-exponential function approach to the q-deformation of time evolution -- 3.7 Alternative realizations of discrete time evolutions and stationary solutions -- 4. Discrete Schrodinger Equations. Typical Examples -- 4.1 The isotropic harmonic oscillator on the lattice -- 4.2 Hopping particle in a linear potential -- 4.3 The Coulomb potential on the Bethe-lattice -- 4.4 The discrete s-wave description of the Coulombproblem -- 4.5 The Maryland class of potentials -- 4.6 The relativistic quasipotential approach to the Coulomb-problem -- 4.7 The infinite square well -- 4.8 Other discrete systems.

5. Discrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl Algebras -- 5.1 Lie algebraic approach to the discretization of differential equations -- 5.2 Describing exactly and quasi-exactly solvable systems -- 5.3 The discrete analog of the harmonic oscillator -- 5.4 Applying the factorization method -- 5.5 The discrete analog of the radial Coulomb-problem -- 5.6 The discrete analog of the isotropic harmonic oscillator -- 5.7 Realizations of Heisenberg-Weyl commutation relations -- 6. Hopping Hamiltonians. Electrons in Electric Field -- 6.1 Periodic and fixed boundary conditions -- 6.2 Density of states and Lyapunov exponents -- 6.3 The localization length: an illustrative example -- 6.4 Delocalization effects -- 6.5 The influence of a time dependent electric field -- 6.6 Discretized time and dynamic localization -- 6.7 Extrapolations towards more general modulations -- 6.8 The derivation of the exact wavefunction revisited -- 6.9 Time discretization approach to the minimum of the MSD -- 6.10 Other methods to the derivation of the DLC -- 6.11 Rectangular wave fields and other generalizations -- 6.12 Wannier-Stark ladders -- 6.13 Quasi-energy approach to DLC's -- 6.14 The quasi-energy description of dc-ac fields -- 6.15 Establishing currents in terms of the Boltzmann equation -- 7. Tight Binding Descriptions in the Presence of the Magnetic Field -- 7.1 The influence of the nearest and next nearest neighbors -- 7.2 Transition to the wavevector representation -- 7.3 The secular equation -- 7.4 The Q = 2 integral quantum Hall effect -- 7.5 Duality properties -- 7.6 Tight binding descriptions with inter-band couplings -- 7.7 Concrete single-band equations and classical realizations -- 8. The Harper-Equation and Electrons on the 1D Ring -- 8.1 The usual derivation of the Harper-equation -- 8.2 The transfer matrix.

8.3 The derivation of Δ-dependent energy polynomials -- 8.4 Deriving Δ-dependent DOS-evaluations -- 8.5 Numerical DOS-studies -- 8.6 Thermodynamic and transport properties -- 8.7 The 1D ring threaded by a time dependent magnetic flux -- 8.8 The tight binding description of electrons on the 1D ring -- 8.9 The persistent current for the electrons on the 1D discretized ring at T =0 -- 9. The q-Symmetrized Harper Equation -- 9.1 The derivation of the generalized qSHE -- 9.2 The three term recurrence relation -- 9.3 Symmetry properties -- 9.4 The SLq (2)-symmetry of the q SHE -- 9.5 Magnetic translations -- 9.6 The SUq(2)-symmetry of the usual Harper Hamiltonian -- 9.7 Commutation relations concerning magnetic translation operators and the Hamiltonian -- 10. Quantum Oscillations and Interference Effects in Nanodevices -- 10.1 The derivation of generalized formulae to the total persistent current in terms of Fourier-series -- 10.2 The discretized Aharonov-Bohm ring with attached leads -- 10.3 Quantum wire attached to a chain of quantum dots -- 10.4 Quantum oscillations in multichain nanorings -- 10.5 Quantum LC-circuits with a time-dependent external source -- 10.6 Dynamic localization effects in L-ring circuits -- 10.7 Double quantum dot systems attached to leads -- 11. Conclusions -- 11.1 Further perspectives -- Appendix A Dealing with polynomials of a discrete variable -- Appendix B The functional Bethe-ansatz solution -- Bibliography -- Index.