由于约束而导致的可分性的分解，提交给数学物理报告
Breakdown of separability due to confinement, submitted to Report on mathematical physics
摘要：
研究了二维位形空间$S$中的两粒子简单系统。假设了在$S$中将系统的与时间无关的薛定谔方程分裂为两个分离的一维单体薛定谔方程的可能性。在本文中，我们专注于后者的属性是如何反击施加这样的边界条件限制在有限区域的$S$和/或限制联合坐标概率密度产生的相对坐标的符号不变条件（不可穿透条件）。我们的研究表明，减少的问题下审查到一个单一的粒子生活在一个有限的域的二维构型空间。这些一般的想法说明引入坐标$X_c$和$x$的质心的两个粒子和相关的相对运动，分别。然后通过研究适当的格林函数和$X_c$和$x$的协方差的时间演化来分析约束和不可穿透性的影响。此外，本文还应用格林函数的Jacobi $θ_3$ -函数表达式计算了单粒子在正方形、菱形、三角形和矩形内的状态。所有的结果都通过实例说明。
英文摘要：
A simple system of two particles in a bidimensional configurational space $S$ is studied. The possibility of breaking in $S$ the time independent Schrödinger equation of the system into two separated one-dimensional one-body Schrödinger equations is assumed. In this paper, we focus on how the latter property is countered by imposing such boundary conditions as confinement in a limited region of $S$ and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate (an impenetrability condition). Our investigation demonstrates the reducibility of the problem under scrutiny into that of a single particle living in a limited domain of its bidimensional configurational space. These general ideas are illustrated introducing the coordinates $X_c$ and $x$ of the center of mass of two particles and of the associated relative motion, respectively. The effects of the confinement and the impenetrability are then analyzed by studying with the help of an appropriate Green's function and the time evolution of the covariance of $X_c$ and $x$. Moreover, to calculate the state of the single particle constrained within a square, a rhombus, a triangle and a rectangle the Green's function expression in terms of Jacobi $θ_3$-function is applied. All the results are illustrated by examples.
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