We aim to compute the discrete energy spectrum for two-body scattering in a three-dimensional box under periodic boundary conditions. The spectrum in the center of mass is obtained by solving the Schödinger equation in a test potential using the Fourier basis. The focus is on how to project the spectrum into the various irreducible representations of the symmetry groups of the box. Four examples are given to show how the infinite-volume spectrum (including both bound and scattering states) is resolved in cubic or elongated boxes, and in systems with integer or half-integer total spin. Such a demonstration is a crucial step in relating the discrete spectrum in the box to the infinite-volume scattering phaseshifts via the Lüscher method.

中文翻译：

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周期盒内两粒子散射的能谱

我们的目标是计算周期性边界条件下三维盒子中二维散射的离散能谱。质心光谱是通过使用傅里叶基在测试电位中求解薛定谔方程获得的。重点是如何将光谱投影到盒子对称群的各种不可约表示中。给出了四个例子来说明无限体积光谱（包括束缚态和散射态）如何在立方或拉长的盒子中以及在具有整数或半整数总自旋的系统中解析。这样的演示是通过 Lüscher 方法将盒子中的离散光谱与无限体积散射相移联系起来的关键步骤。