Eigenvalue asymptotics for the one-particle density matrix and one-particle kinetic energy density operator

Abstract

Let $\psi (\mathbf{x})$, $\mathbf{x}\in {\mathbb{R}}^{3N}$, be an eigenfunction of the $N$-particle atomic Schrödinger operator. We consider the one-particle density matrix $\gamma (x,y)$ and one-particle kinetic energy density $\varkappa (x,y)$, $x,y\in {\mathbb{R}}^3$, associated with the eigenfunction $\psi$. Both functions play a central role in quantum chemistry computations of atomic and molecular bound states: the knowledge of the eigenvalue behaviour of the integral operators $\mathsf{\Gamma }$ and $\mathsf{K}$ with kernels $\gamma (x,y)$ and $\varkappa (x,y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues ${\lambda }_k(\mathsf{\Gamma })>0$ and ${\lambda }_k(\mathsf{K})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,λ_k({\sfΓ}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,λ_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $\psi$. These asymptotics are determined by the singularities of the function $\psi$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $\psi$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $\psi$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues ${\lambda }_k(\mathsf{\Gamma })$ and ${\lambda }_k(\mathsf{K})$. The asymptotic formulas take the form \[ \lim_{k\to \infty} k^{\frac{10}{3}} \,λ_k({\sfΓ}) = \big(A_{asym}\big)^{\frac{10}{3}},\quad \lim_{k\to \infty} k^{\frac{8}{3}} \,λ_k({\sf K}) = \big(B_{asym}\big)^{\frac{8}{3}}, \] where $A_{asym}$ and $B_{asym}$ are non-negative constants given explicitly in terms of the gradient of $\psi$.