Preface ....................................................... vii
1 Lattice Structures and Discretizations ....................... 1
1.1 Discrete derivatives .................................... 1
1.2 The Jackson derivative .................................. 3
1.3 The g-integral .......................................... 6
1.4 Generalized q-hypergeometric functions .................. 7
1.5 The discrete space-time: a short retrospect ............. 9
1.6 Quick inspection of q-deformed Schrödinger equations ... 13
1.7 Orthogonal polynomials of hypergeometric type on
the discrete space ..................................... 14
2 Periodic Quasiperiodic and Confinement Potentials ........... 17
2.1 Short derivation of the В loch-theorem ................. 17
2.2 The derivation of energy-band structures ............... 19
2.3 Direct and reciprocal lattices ......................... 22
2.4 Quasiperiodic potentials ............................... 25
2.5 A shorthand presentation of the elliptic
Lame-equation .......................................... 27
2.6 Quantum dot potentials ................................. 28
2.7 Quantum ring potentials ................................ 31
2.8 Persistent currents and magnetizations ................. 32
2.9 The derivation of the total persistent current for
electrons on the ID ring at T = 0 ...................... 35
2.10 Circular currents ..................................... 37
3 Time Discretization Schemes ................................. 41
3.1 Discretized time evolutions of coordinate and
momentum observables ................................... 42
3.2 Time independent Hamiltonians of hyperbolic type ....... 43
3.3 Time independent Hamiltonians of elliptic type ......... 45
3.4 The derivation of matrix elements ...................... 46
3.5 Finite difference Liouville-von Neumann equations and
"elementary" time scales ............................... 48
3.6 The q-exponential function approach to the
q-deformation of time evolution ........................ 50
3.7 Alternative realizations of discrete time evolutions
and stationary solutions ............................... 55
4 Discrete Schrodinger Equations. Typical Examples ............ 57
4.1 The isotropic harmonic oscillator on the lattice ....... 58
4.2 Hopping particle in a linear potential ................. 61
4.3 The Coulomb potential on the Bethe-lattice ............. 65
4.4 The discrete s-wave description of the
Coulomb-problem ........................................ 66
4.5 The Maryland class of potentials ....................... 69
4.6 The relativistic quasipotential approach to the
Coulomb-problem ........................................ 73
4.7 The infinite square well ............................... 75
4.8 Other discrete systems ................................. 76
5 Discrete Analogs and Lie-Algebraic Discretizations.
Realizations of Heisenberg-Weyl Algebras .................... 79
5.1 Lie algebraic approach to the discretization of
differential equations ................................. 80
5.2 Describing exactly and quasi-exactly solvable
systems ................................................ 82
5.3 The discrete analog of the harmonic oscillator ......... 84
5.4 Applying the factorization method ...................... 87
5.5 The discrete analog of the radial Coulomb-problem ...... 89
5.6 The discrete analog of the isotropic harmonic
oscillator ............................................. 93
5.7 Realizations of Heisenberg-Weyl commutation
relations .............................................. 95
6 Hopping Hamiltonians. Electrons in Electric Field ........... 99
6.1 Periodic and fixed boundary conditions ................ 101
6.2 Density of states and Lyapunov exponents .............. 103
6.3 The localization length: an illustrative example ...... 105
6.4 Derealization effects ................................. 107
6.5 The influence of a time dependent electric field ...... 108
6.6 Discretized time and dynamic localization ............. 111
6.7 Extrapolations towards more general modulations ....... 114
6.8 The derivation of the exact wavefunction revisited .... 116
6.9 Time discretization approach to the minimum of
the MSD ............................................... 118
6.10 Other methods to the derivation of the DLC ............ 120
6.11 Rectangular wave fields and other generalizations ..... 122
6.12 Wannier-Stark ladders ................................. 125
6.13 Quasi-energy approach to DLC's ........................ 126
6.14 The quasi-energy description of dc-ac fields .......... 129
6.15 Establishing currents in terms of the Boltzmann
equation .............................................. 131
7 Tight Binding Descriptions in the Presence of the
Magnetic Field ............................................. 133
7.1 The influence of the nearest and next nearest
neighbors ............................................. 134
7.2 Transition to the wavevector representation ........... 136
7.3 The secular equation .................................. 138
7.4 The Q = 2 integral quantum Hall effect ................ 140
7.5 Duality properties .................................... 142
7.6 Tight binding descriptions with inter-band
couplings ............................................. 143
7.7 Concrete single-band equations and classical
realizations .......................................... 147
8 The Harper-Equation and Electrons on the ID Ring ........... 151
8.1 The usual derivation of the Harper-equation ........... 152
8.2 The transfer matrix ................................... 153
8.3 The derivation of Δ-dependent energy polynomials ...... 155
8.4 Deriving Δ-dependent DOS-evaluations .................. 157
8.5 Numerical DOS-studies ................................. 160
8.6 Thermodynamic and transport properties ................ 161
8.7 The ID ring threaded by a time dependent magnetic
flux .................................................. 167
8.8 The tight binding description of electrons on
the 1D ring ........................................... 170
8.9 The persistent current for the electrons on the ID
discretized ring at T = 0 ............................. 172
9 The q-Symmetrized Harper Equation .......................... 175
9.1 The derivation of the generalized qSHE ................ 175
9.2 The three term recurrence relation .................... 178
9.3 Symmetry properties ................................... 181
9.4 The SLq (2)-symmetry of the q SHE ..................... 184
9.5 Magnetic translations ................................. 188
9.6 The SUq (2)-symmetry of the usual Harper
Hamiltonian ........................................... 190
9.7 Commutation relations concerning magnetic
translation operators and the Hamiltonian ............. 192
10 Quantum Oscillations and Interference Effects in
Nano devices ............................................... 195
10.1 The derivation of generalized formulae to the total
persistent current in terms of Fourier-series ......... 196
10.2 The discretized Aharonov-Bohm ring with attached
leads ................................................. 199
10.3 Quantum wire attached to a chain of quantum dots ...... 207
10.4 Quantum oscillations in multichain nanorings .......... 210
10.5 Quantum LC-circuits with a time-dependent external
source ................................................ 215
10.6 Dynamic localization effects in L-ring circuits ....... 219
10.7 Double quantum dot systems attached to leads .......... 220
11 Conclusions ................................................ 225
11.1 Further perspectives .................................. 228
Appendix A. Dealing with polynomials of a discrete variable ... 231
Appendix В. The functional Bethe-ansatz solution .............. 237
Bibliography .................................................. 241
Index ......................................................... 259
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