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arXiv:2207.03963 · Probability and Mathematical Physics

The diagonal behaviour of the one-particle Coulombic density matrix

Peter Hearnshaw, Alexander V. Sobolev

Published Nov 29, 2023• 6 citations
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Abstract

We obtain bounds for all derivatives of the non-relativistic Coulombic one-particle density matrix $γ(x, y)$ near the diagonal $x = y$.

DOI
10.2140/pmp.2023.4.935

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We obtain bounds for all derivatives of the non-relativistic Coulombic one-particle density matrix γ(x,y)γ(x, y)γ(x,y) near the diagonal x=yx = yx=y.
It is shown that the eigenvalues λk,k=1,2,…,λ_k, k=1, 2, \dots,λk​,k=1,2,…, of the one-particle density matrix satisfy the bound λk≤Ck−8/3λ_k\le C k^{-8/3}λk​≤Ck−8/3 with a positive constant CCC.
Analytical Solutions of One-Dimensional (1D1\mathcal{D}1D) Potentials for Spin-0 Particles via the Feshbach-Villars Formalism
We present a unified analytical and numerical study of the one-dimensional Feshbach--Villars (FV) equation for spin-0 particles in the presence of several representative external potentials. Starting from the FV formulation of the Klein--Gordon equation, we derive the corresponding one-dimensional master equation and analyse its solutions for Coulomb, power-exponential, Cornell, Pöschl--Teller, and Woods--Saxon interactions. For the singular Coulomb and Cornell cases, a Loudon-type cutoff regularisation is implemented on the full line, allowing a mathematically controlled treatment of the origin and an explicit classification of the states by parity. The Coulomb problem exhibits the expected near-degenerate even--odd structure in the cutoff limit, while the Cornell potential combines short-distance Coulomb behaviour with long-distance confinement and produces a finite set of bound states for fixed parameters. The power-exponential potential with p=1p=1p=1 is reduced to a Whittaker-type equation and yields an intrinsically relativistic spectrum with no standard Schrödinger bound-state limit in the parameter regime considered. For the smooth short-range Pöschl--Teller and Woods-Saxon potentials, the FV formalism reveals, respectively, the effects of definite parity and spatial asymmetry on the spectrum, wave functions, and particle--antiparticle mixing. In all cases, we reconstruct the full FV spinor, analyse the associated charge density, and compare the relativistic behaviour with the corresponding non-relativistic expectations whenever such a limit exists. The results provide a coherent set of analytical and numerical benchmarks for relativistic scalar bound states in one dimension.
The non-vanishing of the natural orbital (NO) occupation numbers of the one-particle density matrix of many-body systems has important consequences for the existence of a density matrix-potential mapping for nonlocal potentials in reduced density matrix functional theory and for the validity of the extended Koopmans' theorem. On the basis of Weyl's theorem we give a connection between the differentiability properties of the ground state wavefunction and the rate at which the natural occupations approach zero when ordered as a descending series. We show, in particular, that the presence of a Coulomb cusp in the wavefunction leads, in general, to a power law decay of the natural occupations, whereas infinitely differentiable wavefunctions typically have natural occupations that decay exponentially. We analyze for a number of explicit examples of two-particle systems that in case the wavefunction is non-analytic at its spatial diagonal (for instance, due to the presence of a Coulomb cusp) the natural orbital occupations are non-vanishing. We further derive a more general criterium for the non-vanishing of NO occupations for two-particle wavefunctions with a certain separability structure. On the basis of this criterium we show that for a two-particle system of harmonically confined electrons with a Coulombic interaction (the so-called Hookium) the natural orbital occupations never vanish.