Low-Dimensional Nanoscale Systems on Discrete Spaces

Contents......Page 10
Preface......Page 8
1.1 Discrete derivatives......Page 16
1.2 The Jackson derivative......Page 18
1.3 The q-integral......Page 21
1.4 Generalized q-hypergeometric functions......Page 22
1.5 The discrete space-time: a short retrospect......Page 24
1.6 Quick inspection of q-deformed Schrodinger equations......Page 28
1.7 Orthogonal polynomials of hypergeometric type on the discrete space......Page 29
2.1 Short derivation of the Bloch-theorem......Page 32
2.2 The derivation of energy-band structures......Page 34
2.3 Direct and reciprocal lattices......Page 37
2.4 Quasiperiodic potentials......Page 40
2.5 A shorthand presentation of the elliptic Lame-equation......Page 42
2.6 Quantum dot potentials......Page 43
2.7 Quantum ring potentials......Page 46
2.8 Persistent currents and magnetizations......Page 47
2.9 The derivation of the total persistent current for electrons on the 1D ring at T =0......Page 50
2.10 Circular currents......Page 52
3. Time Discretization Schemes......Page 56
3.1 Discretized time evolutions of coordinate and momentum observables......Page 57
3.2 Time independent Hamiltonians of hyperbolic type......Page 58
3.3 Time independent Hamiltonians of elliptic type......Page 60
3.4 The derivation of matrix elements......Page 61
3.5 Finite difference Liouville-von Neumann equations and “elementary” time scales......Page 63
3.6 The q-exponential function approach to the q-deformation of time evolution......Page 65
3.7 Alternative realizations of discrete time evolutions and stationary solutions......Page 70
4. Discrete Schrodinger Equations. Typical Examples......Page 72
4.1 The isotropic harmonic oscillator on the lattice......Page 73
4.2 Hopping particle in a linear potential......Page 76
4.3 The Coulomb potential on the Bethe-lattice......Page 80
4.4 The discrete s-wave description of the Coulombproblem......Page 81
4.5 The Maryland class of potentials......Page 84
4.6 The relativistic quasipotential approach to the Coulomb-problem......Page 88
4.7 The infinite square well......Page 90
4.8 Other discrete systems......Page 91
5. Discrete Analogs and Lie-Algebraic Discretizations. Realizations of Heisenberg-Weyl Algebras......Page 94
5.1 Lie algebraic approach to the discretization of differential equations......Page 95
5.2 Describing exactly and quasi-exactly solvable systems......Page 97
5.3 The discrete analog of the harmonic oscillator......Page 99
5.4 Applying the factorization method......Page 102
5.5 The discrete analog of the radial Coulomb-problem......Page 104
5.6 The discrete analog of the isotropic harmonic oscillator......Page 108
5.7 Realizations of Heisenberg-Weyl commutation relations......Page 110
6. Hopping Hamiltonians. Electrons in Electric Field......Page 114
6.1 Periodic and fixed boundary conditions......Page 116
6.2 Density of states and Lyapunov exponents......Page 118
6.3 The localization length: an illustrative example......Page 120
6.4 Delocalization effects......Page 122
6.5 The influence of a time dependent electric field......Page 123
6.6 Discretized time and dynamic localization......Page 126
6.7 Extrapolations towards more general modulations......Page 129
6.8 The derivation of the exact wavefunction revisited......Page 131
6.9 Time discretization approach to the minimum of the MSD......Page 133
6.10 Other methods to the derivation of the DLC......Page 135
6.11 Rectangular wave fields and other generalizations......Page 137
6.12 Wannier-Stark ladders......Page 140
6.13 Quasi-energy approach to DLC’s......Page 141
6.14 The quasi-energy description of dc-ac fields......Page 144
6.15 Establishing currents in terms of the Boltzmann equation......Page 146
7. Tight Binding Descriptions in the Presence of the Magnetic Field......Page 148
7.1 The influence of the nearest and next nearest neighbors......Page 149
7.2 Transition to the wavevector representation......Page 151
7.3 The secular equation......Page 153
7.4 The Q = 2 integral quantum Hall effect......Page 155
7.5 Duality properties......Page 157
7.6 Tight binding descriptions with inter-band couplings......Page 158
7.7 Concrete single-band equations and classical realizations......Page 162
8. The Harper-Equation and Electrons on the 1D Ring......Page 166
8.1 The usual derivation of the Harper-equation......Page 167
8.2 The transfer matrix......Page 168
8.3 The derivation of Δ-dependent energy polynomials......Page 170
8.4 Deriving Δ-dependent DOS-evaluations......Page 172
8.5 Numerical DOS-studies......Page 175
8.6 Thermodynamic and transport properties......Page 176
8.7 The 1D ring threaded by a time dependent magnetic flux......Page 182
8.8 The tight binding description of electrons on the 1D ring......Page 185
8.9 The persistent current for the electrons on the 1D discretized ring at T =0......Page 187
9.1 The derivation of the generalized qSHE......Page 190
9.2 The three term recurrence relation......Page 193
9.3 Symmetry properties......Page 196
9.4 The SLq (2)-symmetry of the q SHE......Page 199
9.5 Magnetic translations......Page 203
9.6 The SUq(2)-symmetry of the usual Harper Hamiltonian......Page 205
9.7 Commutation relations concerning magnetic translation operators and the Hamiltonian......Page 207
10. Quantum Oscillations and Interference Effects in Nanodevices......Page 210
10.1 The derivation of generalized formulae to the total persistent current in terms of Fourier-series......Page 211
10.2 The discretized Aharonov-Bohm ring with attached leads......Page 214
10.3 Quantum wire attached to a chain of quantum dots......Page 222
10.4 Quantum oscillations in multichain nanorings......Page 225
10.5 Quantum LC-circuits with a time-dependent external source......Page 230
10.6 Dynamic localization effects in L-ring circuits......Page 234
10.7 Double quantum dot systems attached to leads......Page 235
11. Conclusions......Page 240
11.1 Further perspectives......Page 243
Appendix A Dealing with polynomials of a discrete variable......Page 246
Appendix B The functional Bethe-ansatz solution......Page 252
Bibliography......Page 256
Index......Page 274