We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo-Swiech cost. Moreover, by adapting a construction from the study of exchangeable processes in probability theory, we prove that the corresponding $N$-body OT problem is well approximated by the infinite-body problem.
To our class belongs the Coulomb cost which arises in many-electron quantum mechanics. The optimal cost of the Coulombic N-body OT problem as a function of the one-body marginal density is known in the physics and quantum chemistry literature under the name SCE functional, and arises naturally as the semiclassical limit of the celebrated Hohenberg-Kohn functional. Our results imply that in the inhomogeneous high-density limit (i.e. $N\to\infty$ with arbitrary fixed inhomogeneity profile $ρ/N$), the SCE functional converges to the mean field functional.
We also present reformulations of the infinite-body and N-body OT problems as two-body OT problems with representability constraints and give a dual characterization of representable two-body measures which parallels an analogous result by Kummer on quantum representability of two-body density matrices.
Completeness of the Coulomb scattering wave functions
自伴哈密顿量是量子力学的基本组成部分，其本征函数的完备性在核反应和原子核结构理论中起着重要的作用。然而，到目前为止，还没有一个正式的证明的两体哈密顿量的库仑相互作用的本征函数的完整性。本文给出了排斥非屏蔽库仑势的两体库仑散射波函数完备性的第一个形式证明。为了证明完备性，我们使用牛顿方法[R。Newton，J. Math Phys.，1，319（1960）]。证明允许我们声称，本征函数的两体哈密顿与由排斥库仑加短程（核）势的总和给出的潜力也形成一个完整的集合。它还允许一个扩展的Berggren的方法修改的一套完整的本征函数，包括带电粒子的共振。我们还证明了共振伽莫夫函数与库仑尾可以正则化使用Zel'dovich的正则化方法。
General Decomposition of Radial Functions on R^n and Applications to N-Body Quantum Systems
我们提出了一个推广的库仑势的Festhiman-de la Llave分解的任意径向函数$V$上的$R^n$到零在无穷远。这种广义分解可用于将以前关于具有库仑相互作用的$N$ -体量子系统的结果推广到更一般的相互作用类。作为这样的应用程序的一个例子，我们推导出高密度渐近的基态能量与汤川相互作用的热力学极限，使用相关估计格拉夫和Solovej。
