Density-potential mappings in quantum dynamics

Abstract

In a recent letter [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed point problem. This idea was used to generalize the existence and uniqueness theorems underlying time-dependent density functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case as it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that in the most physically relevant cases the fixed point procedure converges. This is further demonstrated with an example.

20 pages, 8 figures, 3 tables

References in corpus (5)

• Real-time electron dynamics with exact-exchange time-dependent density-functional theory
• Global fixed point proof of time-dependent density-functional theory
• Time-dependent V-representability on lattice systems
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Cited by in corpus (21)

• Kohn-Sham Approach to Quantum Electrodynamical Density Functional Theory: Exact Time-Dependent Effective Potentials in Real Space
• A brief compendium of time-dependent density-functional theory
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• Time-dependent density functional theory on a lattice
• Local Control and v-Representability of Correlated Quantum Dynamics
• Time-Dependent Density Functional Theory Beyond Kohn-Sham Slater Determinants
• Interacting fermions in 1D disordered lattices: Exploring localization and transport properties with lattice density-functional theories
• Exact Density-Functionals with Initial-State Dependence and Memory
• Coulomb potentials and Taylor expansions in Time-Dependent Density Functional Theory
• A reformulation of time-dependent Kohn-Sham theory in terms of the second time derivative of the density
• A Snapshot of Time-Dependent Density-Functional Theory
• Computational complexity of time-dependent density functional theory
• Solver for the electronic V-representation problem of time-dependent density functional theory
• Time-dependent density-functional theory for strongly interacting electrons
• Foundations of stochastic time-dependent current-density functional theory for open quantum systems: Potential pitfalls and rigorous results
• Non-existence of Taylor expansion in time due to cusps
• Invertibility of the retarded response functions for initial mixed states: application to one-body reduced density matrix functional theory
• The density gradient expansion of correlation functions
• Invertibility of retarded response functions for Laplace transformable potentials: application to one-body reduced density matrix functional theory
• On the Action Formalism of Time-dependent Density-functional Theory
• Functional differentiability in time-dependent quantum mechanics