Existence of propagators for time-dependent Coulomb-like potentials

Abstract

We prove existence of propagators for a time dependent Schrödinger equation with a new class of softened Coulomb potentials, which we allow to be time dependent, in the context of time dependent density functional theory. We compute explicitly the Fourier transform of these new potentials, and provide an alternative proof for the Fourier transform of the Coulomb potential using distribution theory. Finally we show the new potentials are dilatation analytic, and so the spectrum of the corresponding Hamiltonian can be fully characterized.

Revisiting the energy spectrum of the radial screened Coulomb potential

“…A new type of radial screened Coulomb potential (RSCP) was recently proposed by Stachura [1] to demonstrate the existence of propagators for the time-dependent Schrödinger equation of a general N-particle system within the context of time-dependent density functional theory [2]. The potential is expressed in a compact form V RSCP (r) = − e − C/ r /r (where C is the radial screening parameter), and it recovers the Coulomb potential in the limit of C = 0.…”

“…On the other hand, the RSCP can be considered as a special type of softened Coulomb potential with an exponentially decaying amplitude in the far asymptotic region, which manifests its potential usefulness in the simulation of reduced-dimensionality quantum systems, e.g., atoms or molecules exposed in intense laser fields [4][5][6][7][8], and spatially confined systems with strong correlations [9][10][11][12]. Some mathematical properties of the RSCP, e.g., the dilatation analyticity and Fourier analysis, have also attracted increasing interest in recent years [1,13]. In view of the extensive application of the conventional screened Coulomb potential in chemical physics [14], atomic and plasma physics [15][16][17][18], and condensed matter systems [19][20][21], the newly proposed RSCP may reveal its potential usefulness in these active areas.…”