Eigenvalue asymptotics for the one-particle density matrix and one-particle kinetic energy density operator
Let $ψ({\mathbf x})$, ${\mathbf x} \in{\mathbb R}^{3N}$, be an eigenfunction of the $N$-particle atomic Schrödinger operator. We consider the one-particle density matrix $γ(x, y)$ and one-particle kinetic energy density $\varkappa(x, y)$, $x, y\in {\mathbb R}^3$, associated with the eigenfunction $ψ$. 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 ${\sfΓ}$ and ${\sf{K}}$ with kernels $γ(x, y)$ and $\varkappa(x, y)$ serves to estimate the errors due to finite-dimensional approximations. We find the following asymptotic formulas for their eigenvalues $λ_k({\sfΓ})>0$ and $λ_k({\sf{K}})>0$: \[ \lim_{k\to \infty} k^{\frac{8}{3}} \,λ_k({\sfΓ}) = A^{\frac{8}{3}},\quad \lim_{k\to \infty} k^2\,λ_k({\sf{K}}) = B^2, \] where $A$ and $B$ are non-negative constants given explicitly in terms of the eigenfunction $ψ$. These asymptotics are determined by the singularities of the function $ψ$ at pair coalescence points of the particles. To identify and isolate these singularities we use some recent regularity results for $ψ$. At the last step we apply Birman-Solomyak spectral asymptotics results for pseudodifferential operators with homogeneous symbols. In the special case where the eigenfunction $ψ$ is totally antisymmetric, it exhibits enhanced regularity, which leads to a faster decay of the eigenvalues $λ_k(\sfΓ)$ and $λ_k(\sf{K})$. 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 $ψ$.