论文标题
关于二维棕色 - 雷文霍尔操作员的稳定性和光谱特性
On the stability and spectral properties of the two-dimensional Brown-Ravenhall operator with a short-range potential
论文作者
论文摘要
最初提出了棕色 - 荷兰霍尔操作员，作为通过库仑潜能来描述费米式特性相互作用并受到相对论的替代方案。该操作员的定义是根据相关的Dirac运算符以及对自由DIRAC运算符正频谱子空间的投影定义的。在本文中，我们建议在二维中分析Brown-Ravenhall操作员的修改版本。更具体地说，我们考虑使用Besl-Macdonald功能（也称为$ k_0 $ - 电势）给出的短距离潜力的棕色 - 雷文霍尔操作员，使用Foldy-Wouthuysen统一转换。最初，我们证明，当耦合常数低于指定的临界值（也称为稳定性）时，二维棕色 - 雷文霍尔运算符具有$ k_0 $ - 优势的二维运算符。该模型的一个主要特征是，即使耦合常数高于指定的临界值，它也不会停止在下面的边界。我们还研究了该操作员的光谱的性质，特别是必需频谱的位置，以及特征值的存在，这些特征值要么是从本质频谱中分离出来的，要么嵌入其中。
The Brown-Ravenhall operator was initially proposed as an alternative to describe the fermion-fermion interaction via Coulomb potential and subject to relativity. This operator is defined in terms of the associated Dirac operator and the projection onto the positive spectral subspace of the free Dirac operator. In this paper, we propose to analyze a modified version of the Brown-Ravenhall operator in two-dimensions. More specifically, we consider the Brown-Ravenhall operator with a short-range attractive potential given by a Bessel-Macdonald function (also known as $K_0$-potential) using the Foldy-Wouthuysen unitary transformation. Initially, we prove that the two-dimensional Brown-Ravenhall operator with $K_0$-potential is bounded from below when the coupling constant is below a specified critical value (a property also referred to as stability). A major feature of this model is the fact that it does not cease to be bounded below even if the coupling constant is above the specified critical value. We also investigate the nature of the spectrum of this operator, in particular the location of the essential spectrum, and the existence of eigenvalues, which are either isolated from the essential spectrum or embedded in it.
