6533b7d0fe1ef96bd125b7fb
RESEARCH PRODUCT
Density-potential mappings in quantum dynamics
R. Van LeeuwenMichael RuggenthalerMarkus PenzKlaas J. H. Giesbertzsubject
PhysicsQuantum PhysicsCondensed Matter - Materials ScienceSpacetimeta114Quantum dynamicsOperator (physics)Continuous spectrumMaterials Science (cond-mat.mtrl-sci)FOS: Physical sciencesMathematical Physics (math-ph)01 natural sciencesAtomic and Molecular Physics and Optics010305 fluids & plasmas0103 physical sciencesConvergence (routing)Quantum systemApplied mathematicsUniquenessBoundary value problem010306 general physicsQuantum Physics (quant-ph)Mathematical Physicsdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2012-01-01 |