AI 中文总结
该研究推导了单粒子密度矩阵与单粒子动能密度算子的本征值渐近公式，发现完全反对称本征函数会使本征值衰减更快，结果可用于估计量子化学有限维近似误差。
AI 中文摘要
设ψ(x)（x∈ℝ³ᴺ）为N粒子原子薛定谔算子的本征函数，我们考虑与本征函数ψ关联的单粒子密度矩阵γ(x,y)和单粒子动能密度κ(x,y)（x,y∈ℝ³）。这两个函数在原子与分子束缚态的量子化学计算中处于核心地位：积分算子Γ（核为γ(x,y)）与K（核为κ(x,y)）的本征值行为，可用于估计有限维近似带来的误差。我们得到其本征值λₖ(Γ)>0与λₖ(K)>0的渐近公式：当k→∞时，lim k^(8/3)λₖ(Γ)=A^(8/3)，lim k²λₖ(K)=B²，其中A、B为可由本征函数ψ显式给出的非负常数。这些渐近行为由ψ在粒子对聚结点处的奇点决定，为识别并分离这些奇点，我们运用了ψ的最新正则性结果，最后一步应用了带齐次符号的伪微分算子的Birman-Solomyak谱渐近结果。在本征函数ψ为完全反对称的特殊情形下，其正则性增强，导致λₖ(Γ)与λₖ(K)的本征值衰减更快，对应的渐近公式为：当k→∞时，lim k^(10/3)λₖ(Γ)=(Aₐₛᵧₘ)^(10/3)，lim k^(8/3)λₖ(K)=(Bₐₛᵧₘ)^(8/3)，其中Aₐₛᵧₘ、Bₐₛᵧₘ为可由ψ的梯度显式给出的非负常数。
英文摘要
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 $ψ$.
