The jellium model is commonly used in condensed matter physics to study the properties of a two-dimensional electron gas system. Within this approximation, one assumes that electrons move in the presence of a neutralizing background consisting of uniformly spread positive charge. When properties of bulk systems (of infinite size) are studied, shape of the jellium domain is irrelevant. However, the same cannot be said when one is dealing with finite systems of electrons confined in a finite two-dimensional region of space. In such a case, geometry and shape of the jellium background play a role on the overall properties of the system. In this work, we assume that the region where the electrons are confined is represented by a jellium background charge with an elliptical shape. It is shown that, in this case, the Coulomb self-energy of the elliptically shaped region can be exactly calculated in closed analytical form by using suitable mathematical transformations. The results obtained reveal the external influence of geometry/shape on the properties of two-dimensional systems of few electrons confined to a small finite region of space.

中文翻译：

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椭圆果冻背景的形状相关能量

胶冻模型常用于凝聚态物理研究二维电子气系统的性质。在这种近似中，人们假设电子在存在由均匀分布的正电荷组成的中和背景的情况下移动。当研究体系统（无限大小）的特性时，果冻域的形状是无关紧要的。然而，当人们处理限制在有限二维空间区域中的有限电子系统时，情况就不是这样了。在这种情况下，果冻背景的几何形状和形状会影响系统的整体特性。在这项工作中，我们假设电子被限制的区域由椭圆形的果冻背景电荷表示。表明，在这种情况下，椭圆形区域的库仑自能可以通过使用合适的数学变换以闭合解析形式精确计算。获得的结果揭示了几何形状/形状对受限于一个小的有限空间区域的少数电子的二维系统的特性的外部影响。