论文标题
零范围的零范围潜力：结合状态问题
Zero-range potentials for Dirac particles: Bound-state problems
论文作者
论文摘要
$ \ mathbb {r}^{3} $中的狄拉克粒子由$ n \ geqslant1 $绑定的模型显示了零分布的零范围势。粒子与电势之间的相互作用是通过使粒子的双波波函数在电势中心的某些限制条件下建模的。这些条件中的每一个都通过$ 2 \ times2 $ hermitian矩阵（或等效地，一个真实的标量和真实矢量）进行参数，并混合波函数的上部和下部成分。确定粒子的结合状态征素耐药的问题减少到找到某个$ 2N \ times2n $矩阵的确定因素的真实零的问题。由于粒子波函数的较低分量在每个电势中心是逆方奇异的，因此波函数本身是不可正常的。然而，人们可以用属于不同特征力的波函数的特性定义标量伪产物，相对于它是正交的。然后可以将波函数归一化，以使其自峰产品加上一个，负1或零。构建了一组辅助函数，并用于得出粒子矩阵绿色函数的明确表示。为了说明目的，详细研究了两个特定的系统：1）在单个零范围电位的磁场中结合的粒子，2）在两个相同的零范围电势的场中结合的粒子。
A model in which a Dirac particle in $\mathbb{R}^{3}$ is bound by $N\geqslant1$ spatially distributed zero-range potentials is presented. Interactions between the particle and the potentials are modeled by subjecting a particle's bispinor wave function to certain limiting conditions at the potential centers. Each of these conditions is parametrized by a $2\times2$ Hermitian matrix (or, equivalently, a real scalar and a real vector) and mixes the upper and the lower components of the wave function. The problem of determining particle's bound-state eigenenergies is reduced to the problem of finding real zeroes of a determinant of a certain $2N\times2N$ matrix. As the lower component of the particle's wave function is inverse-square singular at each of the potential centers, the wave function itself is not square-integrable. Nevertheless, one can define a scalar pseudo-product with the property that wave functions belonging to different eigenenergies are orthogonal with respect to it. The wave functions may then be normalized so that their self-pseudo-products are plus one, minus one or zero. An auxiliary set of Sturmian functions is constructed and used to derive an explicit representation of particle's matrix Green's function. For illustration purposes, two particular systems are studied in detail: 1) a particle bound in a field of a single zero-range potential, 2) a particle bound in a field of two identical zero-range potentials.
