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arXiv:1103.1983 · Journal of Physics A Mathematical and Theoretical

Domains of time-dependent density-potential mappings

Markus Penz, Michael Ruggenthaler

Published Jul 27, 2011• 9 citations
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Abstract

The key element in time-dependent density functional theory is the one-to-one correspondence between the one-particle density and the external potential. In most approaches, this mapping is transformed into a certain type of Sturm–Liouville problem. Here we give conditions for existence and uniqueness of solutions and construct the weighted Sobolev space they lie in. As a result, the class of v -representable densities is considerably widened with respect to previous work.

DOI
10.1088/1751-8113/44/33/335208

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The key element in time-dependent density functional theory is the one-to-one correspondence between the one-particle density and the external potential. In most approaches, this mapping is transformed into a certain type of Sturm–Liouville problem. Here we give conditions for existence and uniqueness of solutions and construct the weighted Sobolev space they lie in. As a result, the class of v -representable densities is considerably widened with respect to previous work.
This work studies in detail the possibility of defining a one-to-one mapping from charge densities as obtained by the time-dependent Schrödinger equation to external potentials. Such a mapping is provided by the Runge-Gross theorem and lies at the very core of time-dependent density functional theory. After introducing the necessary mathematical concepts, the usual mapping "there" - from potentials to wave functions as solutions to the Schrödinger equation - is revisited paying special attention to Sobolev regularity. This is scrutinised further when the question of functional differentiability of the solution with respect to the potential arises, a concept related to linear response theory. Finally, after a brief introduction to general density functional theory, the mapping "back again" - from densities to potentials thereby inverting the Schrödinger equation for a fixed initial state - is defined. Apart from utilising the original Runge-Gross proof this is achieved through a fixed-point procedure. Both approaches give rise to mathematical issues, previously unresolved, which however could be dealt with to some extent within the framework at hand.
In a recent paper [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 because 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.
In this work we review the mapping from densities to potentials in quantum mechanics, which is the basic building block of time-dependent density-functional theory and the Kohn-Sham construction. We first present detailed conditions such that a mapping from potentials to densities is defined by solving the time-dependent Schrödinger equation. We specifically discuss intricacies connected with the unboundedness of the Hamiltonian and derive the local-force equation. This equation is then used to set up an iterative sequence that determines a potential that generates a specified density via time propagation of an initial state. This fixed-point procedure needs the invertibility of a certain Sturm-Liouville problem, which we discuss for different situations. Based on these considerations we then present a discussion of the famous Runge-Gross theorem which provides a density-potential mapping for time-analytic potentials. Further we give conditions such that the general fixed-point approach is well-defined and converges under certain assumptions. Then the application of such a fixed-point procedure to lattice Hamiltonians is discussed and the numerical realization of the density-potential mapping is shown. We conclude by presenting an extension of the density-potential mapping to include vector-potentials and photons.
We reformulate the uniqueness and existence proofs of time-dependent density-functional theory. The central idea is to restate the fundamental one-to-one correspondence between densities and potentials as a global fixed-point question for potentials on a given time interval. We show that the unique fixed point, i.e. the unique potential generating a given density, is reached as the limiting point of an iterative procedure. The one-to-one correspondence between densities and potentials is a straightforward result provided that the response function of the divergence of the internal forces is bounded. The existence, i.e. the v -representability of a density, can be proven as well provided that the operator norms of the response functions of the members of the iterative sequence of potentials have an upper bound. The densities under consideration have second time-derivatives that are required to satisfy a condition slightly weaker than being square-integrable. This approach avoids the usual restrictions of Taylor-expandability in time of the uniqueness theorem by Runge and Gross ( Phys. Rev. Lett. , 52 (1984) 997 ) and of the existence theorem by van Leeuwen ( Phys. Rev. Lett. , 82 (1999) 3863 ). Owing to its generality, the proof not only answers basic questions in density-functional theory but also has potential implications in other fields of physics.