arXiv subjects

Markus Penz

Publications and source records attributed to Markus Penz.

43 records · Page 3Linked to original sources

arXiv2016

The Density-Potential Mapping in Quantum Dynamics

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.

math-ph↗

arXiv2015

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.

cond-mat.other↗

arXiv2015

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.

math-ph↗

arXiv2011

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.

math-ph↗

arXiv2010

Non-differentiable Bohmian trajectories

A solution $ψ$ to Schrödinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown ψ/mψ\right) .$ In the case of one specific non-differentiable weak solution $Ψ$ we show how Bohmian trajectories can be obtained for $Ψ$ from the trajectories of a sequence $Ψ_{n}\rightarrow Ψ.$ (For any real $t$ the sequence $Ψ_{n}\left( t,\cdot \right) $ converges strongly.) The limiting trajectories no longer need to be differentiable. This suggests a way how Bohmian mechanics might work for arbitrary initial vectors $Ψ$ in the Hilbert space on which the Schrödinger evolution $% Ψ\mapsto e^{-iht}Ψ$ acts.

quant-ph↗

arXiv2008

Fleming's bound for the decay of mixed states

Fleming's 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 $ρ$ there holds the estimate $\Tr(Π\rme^{-\rmi ht}ρ\rme^{\rmi ht}) \geq\cos^{2}((Δh)_ρt) $ for all real $t$ with $(Δh)_ρ| t| \leqπ/2.$ We show that equality either holds for all $t\in\mathbb{R}$ or it does not hold for a single $t$ with $0<(Δh)_ρ| t| \leqπ/2.$ All the density operators saturating the bound for all $t\in\mathbb{R},$ i.e. the mixed intelligent states, are determined.

quant-ph↗

arXiv2005

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 not 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) wave function of maximal backflow are displayed.

quant-ph↗