Eigenvalue Estimates for the Coulombic One-Particle Density Matrix and the Kinetic Energy Density Matrix

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 \(\lambda_k(\gamma)\le C k^{-8/3}\) and \(\lambda_k(\tau)\le C k^{-2}\) 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: \(\lambda_k(\gamma)\le C k^{-10/3}\) and \(\lambda_k(\tau)\le C k^{-8/3}\).

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.