Markus Penz

Existence, uniqueness, and construction of the density-potential mapping in time-dependent density-functional theory

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.

Density-potential mappings in quantum dynamics

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.

A new approach to quantum backflow

We derive some rigorous results concerning the backflow operator introduced by Bracken and Melloy. We show that it is linear bounded, self-adjoint and non-compact. Thus the question is underlined whether the backflow constant is an eigenvalue of the backflow operator. From the position representation of the backflow operator, we obtain a more efficient method to determine the backflow constant. Finally, detailed position probability flow properties of a numerical approximation to the (perhaps improper) wavefunction of maximal backflow are displayed.

Domains of time-dependent density-potential mappings

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.

Functional differentiability in time-dependent quantum mechanics

In this work, we investigate the functional differentiability of the time-dependent many-body wave function and of derived quantities with respect to time-dependent potentials. For properly chosen Banach spaces of potentials and wave functions, Fréchet differentiability is proven. From this follows an estimate for the difference of two solutions to the time-dependent Schrödinger equation that evolve under the influence of different potentials. Such results can be applied directly to the one-particle density and to bounded operators, and present a rigorous formulation of non-equilibrium linear-response theory where the usual Lehmann representation of the linear-response kernel is not valid. Further, the Fréchet differentiability of the wave function provides a new route towards proving basic properties of time-dependent density-functional theory.

Fleming's bound for the decay of mixed states

Flemings inequality is generalized to the decay function of mixed states. We show that for any symmetric Hamiltonian h and for any density operator ρ on a finite-dimensional Hilbert space with the orthogonal projection Π onto the range of ρ the estimate holds for all t with (Δh)ρ|t| ≤ π/2. We show that equality either holds for all or it does not hold for a single t with 0 < (Δh)ρ|t| ≤ π/2. All the density operators saturating the bound for all , i.e. the mixed intelligent states, are determined.

On the existence of effective potentials in time-dependent density functional theory

We investigate the existence and properties of effective potentials in time-dependent density functional theory. We outline conditions for a general solution of the corresponding Sturm–Liouville boundary value problems. We define the set of potentials and v-representable densities, give a proof of existence of the effective potentials under certain restrictions and show the set of v-representable densities to be independent of the interaction.

Arrival time and backflow effect

We contrast the average arrival time at x according to the Bohmian mechanics of one dimensional free Schrodinger evolution with the standard quantum mechanical one. For positive momentum wave functions the first cannot be larger than the second one. Equality holds if and only if the wave function does not lead to position probability backflow through x. This position probability backflow has the least upper bound of approximately 0.04. We describe a numerical method to determine this backflow constant, introduced by Bracken and Melloy, more precisely and we illustrate the approximate wave function of maximal backflow.