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.

Figure 1

Potential-potential mapping $\mathcal{F}$ of Eq. (11) as composition of mappings $\mathcal{P}$ and $\mathcal{V}$.

Figure 2

Density evolving in time from $t=0$ to $T=0.5$.

Figure 3

The four lowest eigenvalues in time from $t=0$ to $T=0.5$ for the Sturm-Liouville eigenvalue problem. The arrows indicate the time of degeneracy of the two lowest-lying eigenfunctions. The constant $1/D\simeq 4.39$ as indicated at time $t\simeq 0.17$.

Figure 4

Lowest eigenfunction in time from $t=0$ to $T=0.5$ for the Sturm-Liouville eigenvalue problem. The lowest-lying eigenfunction at certain points becomes degenerate with the second-lying eigenstate, whereafter they change position in the spectrum.

Figure 5

Simple example illustrating the meaning of operator norm $C[v_0,v_1]$, which corresponds to the steepest tangent on the interval $[v_0,v_1]$.

Figure 6

Potential $v\left(xt\right)$ in time from $t=0$ to $T=0.1$.

Figure 7

Difference between exact potential $v\left(xt\right)$ and iterated potentials $v_1\left(xt\right)$, $v_{200}\left(xt\right)$, and $v_{1000}\left(xt\right)$ in time from $t=0$ to $T=0.1$. Note the change of scale.

Figure 8

The nonnormalized gerade (blue) and ungerade (red) generalized eigenfunctions for $\lambda =1,10,100$.