Eigenvalue estimates for the coulombic one-particle density matrix and the kinetic energy density matrix
Alexander Sobolev Department of Mathematics, University College London, London, United Kingdom
Abstract:
Consider a bound state (an eigenfunction)
$\psi$ of an atom with
$N$
electrons. We study the spectra of the one-particle density matrix
$\gamma$
and the one-particle kinetic energy density matrix
$\tau$ associated
with
$\psi$. The paper contains two results. First, we obtain the bounds
$\lambda_k(\gamma)\leqslant C k^{-8/3}$ and
$\lambda_k(\tau)\leqslant C k^{-2}$
with some positive constants
$C$
that depend explicitly on the eigenfunction
$\psi$. The sharpness
of these bounds is confirmed by the asymptotic results obtained
by the author in earlier papers. The advantage of these bounds over
the ones derived by the author previously is their explicit dependence
on the eigenfunction. Moreover, their new proofs are more elementary and
direct. The second result is new, and it pertains to the case where
the eigenfunction
$\psi$ vanishes at the particle coalescence points.
In particular, this is true for totally antisymmetric
$\psi$.
In this case, the eigenfunction
$\psi$ exhibits enhanced regularity
at the coalescence points, which leads to the faster decay of the eigenvalues:
$\lambda_k(\gamma)\leqslant C k^{-10/3}$ and
$\lambda_k(\tau)\leqslant C k^{-8/3}$.
The proofs rely on estimates for the derivatives of the eigenfunction
$\psi$ that depend explicitly on the distance to the coalescence points.
Some of these estimates are borrowed directly from, and some are derived
using the methods of, a recent paper by S. Fournais and
T. Ø. Sørensen.
Keywords:
multi-particle Schrödinger operator, one-particle density matrix,
eigenvalues,
integral operators.
MSC: 35J10,
47G10,
81Q10 Received: 28.01.2025
Accepted: 25.03.2025
DOI:
10.4213/faa4289
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