论文标题
拉普拉斯操作员，测量浓度，高斯功能和量子力学
The Laplace operator, measure concentration, Gauss functions, and quantum mechanics
论文作者
论文摘要
在本说明中，我们表示电子schrödinger方程的溶液作为较高函数的痕迹。这使得将电子电子相互作用电位解开，但以退化的椭圆运算符的价格取代了较高尺寸的空间上的拉普拉斯操作员。令人惊讶的观察结果是，该操作员可能不会再次被拉普拉斯操作员替代太多损失，越成功，所考虑的系统就越大。这是由于量度效应的集中度与概率理论已知的随机投影定理有很大关系。文本是根据出版物[numer。数学。 146，219--238（2020）]和[Siam J. Matrix肛门。作者的Appl。，43，464--478（2022）]，并根据量子力学的需求调整了这些发现。例如，我们的观察结果可能会在迭代方法中找到用途，这些迭代方法将轨道和宝石的产物总和到相同类型的函数。
We represent in this note the solutions of the electronic Schrödinger equation as traces of higher-dimensional functions. This allows to decouple the electron-electron interaction potential but comes at the price of a degenerate elliptic operator replacing the Laplace operator on the higher-dimensional space. The surprising observation is that this operator can without much loss again be substituted by the Laplace operator, the more successful the larger the system under consideration is. This is due to a concentration of measure effect that has much to do with the random projection theorem known from probability theory. The text is in parts based on the publications [Numer. Math. 146, 219--238 (2020)] and [SIAM J. Matrix Anal. Appl., 43, 464--478 (2022)] of the author and adapts the findings there to the needs of quantum mechanics. Our observations could for example find use in iterative methods that map sums of products of orbitals and geminals onto functions of the same type.
