| Preface | p. vii |
| Lattice Structures and Discretizations | p. 1 |
| Discrete derivatives | p. 1 |
| The Jackson derivative | p. 3 |
| The q-integral | p. 6 |
| Generalized q-hypergeometric functions | p. 7 |
| The discrete space-time: a short retrospect | p. 9 |
| Quick inspection of q-deformed Schrodinger equations | p. 13 |
| Orthogonal polynomials of hypergeometric type on the discrete space | p. 14 |
| Periodic Quasiperiodic and Confinement Potentials | p. 17 |
| Short derivation of the Bloch-theorem | p. 17 |
| The derivation of energy-band structures | p. 19 |
| Direct and reciprocal lattices | p. 22 |
| Quasiperiodic potentials | p. 25 |
| A shorthand presentation of the elliptic Lame-equation | p. 27 |
| Quantum dot potentials | p. 28 |
| Quantum ring potentials | p. 31 |
| Persistent currents and magnetizations | p. 32 |
| The derivation of the total persistent current for electrons on the 1D ring at T = 0 | p. 35 |
| Circular currents | p. 37 |
| Time Discretization Schemes | p. 41 |
| Discretized time evolutions of coordinate and momentum observables | p. 42 |
| Time independent Hamiltonians of hyperbolic type | p. 43 |
| Time independent Hamiltonians of elliptic type | p. 45 |
| The derivation of matrix elements | p. 46 |
| Finite difference Liouville-von Neumann equations and "elementary" time scales | p. 48 |
| The q-exponential function approach to the q-deformation of time evolution | p. 50 |
| Alternative realizations of discrete time evolutions and stationary solutions | p. 55 |
| Discrete Schrodinger Equations. Typical Examples | p. 57 |
| The isotropic harmonic oscillator on the lattice | p. 58 |
| Hopping particle in a linear potential | p. 61 |
| The Coulomb potential on the Bethe-lattice | p. 65 |
| The discrete s-wave description of the Coulomb-problem | p. 66 |
| The Maryland class of potentials | p. 69 |
| The relativistic quasipotential approach to the Coulomb-problem | p. 73 |
| The infinite square well | p. 75 |
| Other discrete systems | p. 76 |
| Discrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl Algebras | p. 79 |
| Lie algebraic approach to the discretization of differential equations | p. 80 |
| Describing exactly and quasi-exactly solvable systems | p. 82 |
| The discrete analog of the harmonic oscillator | p. 84 |
| Applying the factorization method | p. 87 |
| The discrete analog of the radial Coulomb-problem | p. 89 |
| The discrete analog of the isotropic harmonic oscillator | p. 93 |
| Realizations of Heisenberg-Weyl commutation relations | p. 95 |
| Hopping Hamiltonians. Electrons in Electric Field | p. 99 |
| Periodic and fixed boundary conditions | p. 101 |
| Density of states and Lyapunov exponents | p. 103 |
| The localization length: an illustrative example | p. 105 |
| Delocalization effects | p. 107 |
| The influence of a time dependent electric field | p. 108 |
| Discretized time and dynamic localization | p. 111 |
| Extrapolations towards more general modulations | p. 114 |
| The derivation of the exact wavefunction revisited | p. 116 |
| Time discretization approach to the minimum of the MSD | p. 118 |
| Other methods to the derivation of the DLC | p. 120 |
| Rectangular wave fields and other generalizations | p. 122 |
| Wannier-Stark ladders | p. 125 |
| Quasi-energy approach to DLC's | p. 126 |
| The quasi-energy description of dc-ac fields | p. 129 |
| Establishing currents in terms of the Boltzmann equation | p. 131 |
| Tight Binding Descriptions in the Presence of the Magnetic Field | p. 133 |
| The influence of the nearest and next nearest neighbors | p. 134 |
| Transition to the wavevector representation | p. 136 |
| The secular equation | p. 138 |
| The Q = 2 integral quantum Hall effect | p. 140 |
| Duality properties | p. 142 |
| Tight binding descriptions with inter-band couplings | p. 143 |
| Concrete single-band equations and classical realizations | p. 147 |
| The Harper-Equation and Electrons on the 1D Ring | p. 151 |
| The usual derivation of the Harper-equation | p. 152 |
| The transfer matrix | p. 153 |
| The derivation of [Delta]-dependent energy polynomials | p. 155 |
| Deriving [Delta]-dependent DOS-evaluations | p. 157 |
| Numerical DOS-studies | p. 160 |
| Thermodynamic and transport properties | p. 161 |
| The 1D ring threaded by a time dependent magnetic flux | p. 167 |
| The tight binding description of electrons on the 1D ring | p. 170 |
| The persistent current for the electrons on the 1D discretized ring at T = 0 | p. 172 |
| The q-Symmetrized Harper Equation | p. 175 |
| The derivation of the generalized qShe | p. 175 |
| The three term recurrence relation | p. 178 |
| Symmetry properties | p. 181 |
| The SL[subscript q] (2)-symmetry of the q She | p. 184 |
| Magnetic translations | p. 188 |
| The SU[subscript q](2)-symmetry of the usual Harper Hamiltonian | p. 190 |
| Commutation relations concerning magnetic translation operators and the Hamiltonian | p. 192 |
| Quantum Oscillations and Interference Effects in Nanodevices | p. 195 |
| The derivation of generalized formulae to the total persistent current in terms of Fourier-series | p. 196 |
| The discretized Aharonov-Bohm ring with attached leads | p. 199 |
| Quantum wire attached to a chain of quantum dots | p. 207 |
| Quantum oscillations in multichain nanorings | p. 210 |
| Quantum LC-circuits with a time-dependent external source | p. 215 |
| Dynamic localization effects in L-ring circuits | p. 219 |
| Double quantum dot systems attached to leads | p. 220 |
| Conclusions | p. 225 |
| Further perspectives | p. 228 |
| Dealing with polynomials of a discrete variable | p. 231 |
| The functional Bethe-ansatz solution | p. 237 |
| Bibliography | p. 241 |
| Index | p. 259 |
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