What is included with this book?
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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