Abstract
It is proved that the one-particle density matrix gamma (x, y) for multi-particle Coulombic systems is real analytic away from the nuclei and from the diagonal x = y.
Highlights
The objective of the paper is to study analytic properties of the one-particle density matrix for the molecule, consisting of N electrons and N0 nuclei described by the following Schrodinger operator: N
The notation Δk is used for the Laplacian w.r.t. the variable xk
D0 = {(x, y) ∈ R3 × R3 : x = 0, y = 0, x = y}. This result is derived from the following L2-bound on the set D = Dε = {(x, y) ∈ R3 × R3 : |x| > ε, |y| > ε, |x − y| > ε}, ε > 0
Summary
The objective of the paper is to study analytic properties of the one-particle density matrix for the molecule, consisting of N electrons and N0 nuclei described by the following Schrodinger operator: N. Before we state the main result note that regularity of each of the terms in (1.5) can be studied individually It suffices to establish the real analyticity of the function γ(x, y) =. The complete proof of real analyticity of ρ(x) in [10] is more involved It requires the study of various cut-off functions that keep some of the particles “close” to each other, but separate from the rest of them (we call this group of particles the cluster associated with the given cut-off ). 4, we study in detail properties of smooth cut-off functions including the extended cut-offs Φ = Φ(x, y, x), x, y ∈ R3, x ∈ R3N−3, and clusters associated with them.
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